Critical L-values for some quadratic twists of Gross curves

Abstract

Let K=Q(−q)K=\Bbb Q(\sqrt{-q}), where q is a prime congruent to 3 modulo 4. Let A=A(q) denote the Gross curve. Let E=A(−β)E=A^{(-\beta)} denote its quadratic twist, with β=−q\beta=\sqrt{-q}. The curve E is defined over the Hilbert class field H of K. We use Magma to calculate the values L(E/H,1) for all such q's up to some reasonable ranges (different for q≡7 mod 8q\equiv 7 \, \text{mod} \, 8 and q≡3 mod 8q\equiv 3 \, \text{mod} \, 8). All these values are non-zero, and using the Birch and Swinnerton-Dyer conjecture, we can calculate hypothetical orders of \sza(E/H) in these cases. Our calculations extend those given by J. Choi and J. Coates [{\it Iwasawa theory of quadratic twists of X0(49)X_0(49)}, Acta Mathematica Sinica(English Series) {\bf 34} (2017), 19-28] for the case q=7.

Full text

Critical
L
-values for some quadratic twists of
Gross curves
Andrzej Dabrowski, Tomasz Jedrzejak and Lucjan Szymaszkiewicz
Let
K=Q(√−q)
, where
q
is a prime congruent to
3
modulo
4
. Let
A=A(q)
denote the Gross curve [6]. Let
E=A(−β)
denote its quadratic
twist, with
β=√−q
. The curve
E
has the nice explicit equation (see [5],
equation (1.2))
y2=x3−2−43−1(j(OK)1/3x+ 2−53−3(j(OK)−(12)3)1/2,
(1)
where it is understood that, in this equation, we take the real cube root of
j(OK)
, and the square root of
j(OK)−(12)3
lying in the upper half complex
plane. Thus
E
is dened over the Hilbert class eld
H
of
K
. Below we
use Magma [1] to calculate the values
L(E/H, 1)
for all such
q
's up to some
reasonable ranges (dierent for
q≡7
mod
8
and
q≡3
mod
8
). All these
values are non-zero, and using the Birch and Swinnerton-Dyer conjecture, we
can calculate hypothetical orders of
X(E/H )
in these cases. Our calculations
extend those given in [3] for the case
q= 7
.
1 The case
q≡7
mod
8
In this case we know, by a recent result of J. Coates and Y. Li ([5], Theorem
1.3), that
L(E/H, 1) 6= 0
. In the table below we calculate numerically these
values for all such
q
up to
4663
.
Now let us say a few words about the Magma implementation. The
starting source for us was the article by M. Watkins [7], which gives some
numerical examples (or rather hints) how to compute Grossencharacters and
critical L-values (sections 5.4 and 6.1 deals with
Q(√−23)
, but of course
we need to keep track of the eect of twisting). But it was not enough for
us to write an algorithm calculating
L(E/H, 1)
. Watkins [8] corrected our
algorithm (or better, he wrote a new one) and tested for
q= 23
and
79
. It
was a starting point for us to make extensive numerical calculations. The
algorithm uses the fact that
L
-series of an elliptic curve over
H
splits into
1

factors corresponding to Grossencharacters twisted by Hilbert characters and
its conjugates (so it uses the classical Hecke-Deuring theory linking elliptic
curves with
CM
to Grossencharacters, with keeping track of the eect of
twisting). Here are some more details. Assuming
2OK=pp?
, and choosing
the sign of
√−q
so that ord
p((1 −β)) >0
, we can check that
E/H
has good
reduction outside the primes of
H
lying above
p
(see [5]). Moreover, the
Deuring Grossencharacter
ψE/H
of
E/H
is then equal to
ρ◦NH/K
, where
ρ
is the Grossencharacter of
K
with conductor
p2
dened by
ρ(a) = α, ah=αOK, α ≡1
mod
p2.
The algorithm computes the values
L(ρχ, 1)
, where
χ
runs over the characters
of the ideal class group of
K
. Now thanks to the above formula and Deuring's
theory it follows that
L(E/H, 1)
will be given by the product of all the
L(ρχ, 1)
's and their complex conjugates.
Now we know by Iwasawa theory that the Tate-Shafarevich group
X(E/H )
is nite because
L(E/H, 1) 6= 0
. Below we will write down an explicit con-
jectural formula for the order of
X(E/H )
. Let
h
denote the class number of
K
,
m=q−1
4−h
2
, and let
Ω(q)
be a period dened in [6]:
Ω(q) = 1
(2π)m·qh
2Y
0<c<q
(c
q)=1
Γc
q.
The prime
2
splits in
K
, and we write
2OK=pp?
, where we have chosen the
sign of
β=√−q
so that ord
p((1 −β)) >0
. Then
E
is the quadratic twist of
the Gross curve
A/H
by
H(√−β)/H
. Let
{v1, ..., vr}
be the set of primes of
H
lying above
p
, so that
r=h/j
, where
j
is the exact order of the class of
p
in the ideal class group of
K
. It turned out that there are exactly
18
primes
q≤4663
congruent to
7
modulo
8
, for which
r > 1
.
Conjecture 1
#(X(E/H )) = L(E /H, 1)2h+6−2r/(Ω(q)2√q)
.
One easily checks that in case
q= 7
, the formula from Conjecture 1 is
equivalent to (2.11) from [3]. The above conjecture agrees with the Birch and
Swinnerton-Dyer conjecture for
E/H
. In particular, the set of bad primes of
E/H
is precisely
{v1, ..., vr}
, and the factor
2−2r
in the above formula takes
account of the fact that the Tamagawa factor at each of these primes is
4
.
John Coates informed one of us (A. D.) that one should be able to use
the Iwasawa theory being developed in [4] to prove the above conjecture.
Below we use Conjecture 1 to calculate
#(X(E/H ))
(i.e. the analytic
order of
X(E/H )
) for all primes
q
congruent to
7
modulo
8
up to
4663
.
2

q h L(E/H, 1) q#(X(E/H ))
7 1 0.30903153751765917103 1
23 3 0.79196294535428296044 1
31 3 0.35288571505654851763 1
47 5 3.25049251883301426121 3
71 7 0.10920125590289049507 1
79 5 0.02577591231345318312 1
103 5 0.84244014254446144514 13
127 5 0.33138747507581642444 17
151 7 0.00899919291175804982 5
167 11 338.84342541058916626822 2049
191 13 0.07538843930533773444 81
199 9 0.00178784908116291475 9
223 7 0.66858391145992740299 289
239 15 0.02401256252449269664 311
263 13 0.24799355777337639904 1767
271 11 0.00495300516895988511 127
311 19 0.08289319536914465106 12559
359 19 0.00008262935013341212 2057
367 9 0.02393861560648477609 1679
383 17 1058.78512825720370837609 9090067
431 21 5.38876192180481196261 2039928
439 15 0.00002907103487183395 1279
463 7 0.08500423491817571054 11663
479 25 1483.07868786791841546796 1746287691
487 7 0.14694723669623207042 22807
503 21 260759.24737728583571680044 2880463783
599 25 0.00000001076991986883 162285
607 13 0.10795424186869623536 884605
631 13 0.00004385443164140780 44425
647 23 0.00000607641351529086 4925391
719 31 0.00530561250904147645 31646320057
727 13 0.00299892234779012135 1113693
743 21 1498.05565627050935641062 332146468299
751 15 0.00000000896688802629 10512
823 9 0.00102716234866514602 469855
839 33 0.12347121525795507868 9315485111867
863 21 4.28771850132666368851 240267975371
887 29 19477038.03896518654448808873 30785347392739103
911 31 0.00000312260876667640 98895319091
919 19 0.00000000001480555834 36741
967 11 0.09804744238584120050 67762715
983 27 80292384.19994893868777292158 238138622744502833
991 17 0.00004920969572684842 91561037
1031 35 0.06252031110591168794 1595084268489133
1039 23 0.00000369632068956271 1196802971
1063 19 1.67498940035011217030 54030361471
1087 9 0.00390824946108466869 22866381
1103 23 0.00000572944338432493 80104082513
1151 41 0.16613607401358903141 1320694330164429335
1223 35 0.00194039562173745137 9307326019643999
1231 27 0.00000000093497039710 3758406353
1279 23 0.00000000287686176036 1814619877
1303 11 0.00021448455521015721 157522307
1319 45 28941477.30398577348268950039 494146202273056454638285
1327 15 0.00028916732527519815 1982319913
1367 25 28601.01936469536020713962 1839953001047559952
1399 27 0.00000001208663958591 210925423356
1423 9 0.01738234231379547068 1715895524
1439 39 0.00000006661042814296 30644209290657623
1447 23 0.00013884708802240947 811815992737
1471 23 0.00000000029882112555 5374292551
1487 37 6576514753.03121783833652353519 10937565775616401748256581
1511 49 0.00055060129228412947 25127641761490803096975
1543 19 0.00176400518665963757 1034981247929
1559 51 0.87726903612746376846 2738379667079823194690949
1567 15 0.20108355789279651145 1575131870837
1583 33 0.13269489544044152769 15757834130321482863
1607 27 0.55194730530618115640 349749435110817399
1663 17 0.00119021469463015769 513229201123
1759 27 0.00000000101913474126 1344751145077
1783 17 0.00024486558886747311 1885276250643
1823 45 9746102.85279061269213579273 1808382408390942813565214784
1831 19 0.00000000000009270267 880904745
1847 43 1207965.54129407491879223532 4921203932360352431818054853
1871 45 0.00000467957783395582 211817576664143143791049
1879 27 0.00000055405311625279 359706350854836
1951 33 0.00000000000191765783 71833637298811
1999 27 0.00000000047804213103 21632373948999
2039 45 0.00000000367413120950 7999274603520740597625
2063 45 2766645.80955923764536501056 1341376803998421383216338486800
2087 35 164.98809160497598617071 7549276579545660794505221
Continued on next page
3

q h L(E/H, 1) q#(X(E/H ))
2111 49 0.00109168602722010353 4327178311989716140475456427
2143 13 0.01476049903300890724 12322652032019
2207 39 8356.57170711443065175664 5015914871462321771139549955
2239 35 0.00000000000004393777 11915930073079139
2287 29 0.00766729713419227393 6824315445622624657
2311 29 0.00000000187662881760 20015174351579955
2351 63 552470425.54422669425872718713 2816977737852577527309898650097367443659
2383 29 0.00002630161314175636 1162866812268136121
2399 59 0.04141467915645522162 13235014758527507066087807686368973
2423 33 250.16970151756207630008 281045797696507991427800573
2447 37 23.82978667301734246096 2092852385170614860214887843
2503 21 0.02648336374331768867 100742172570243491
2543 35 382.95959917323265302185 10157978602722623458338530496
2551 41 0.00000000000008248960 60637243039930628065
2591 57 0.00000000000238017737 336275233499026658311829296679
2647 15 0.03372248437933683851 6798505360748663
2663 43 822.32375569716777860250 76481720773755781866561562664447
2671 23 0.00000000000003162935 66697260897735
2687 51 194688424.12336279094797547720 17321740551070983627343715625416872652
2711 53 0.37421143332764690649 38629854923911845038305204026873649
2719 41 0.00000000000083400226 1406443138529321307393
2767 21 0.31592061458094422698 3800789493866165232
2791 39 0.00000000000054815909 897017319101036106209
2879 57 0.00000013893708474299 10273597469646245008601935022410689
2887 25 0.00000727449204224937 4682281660959493201
2903 59 469211113283732.46988840553969223592 400977393247118763374214959073343661518898573
2927 31 64.29980720973840221765 27260274502712067357277082151
2999 73 360.95151202061450399689 3943185992249268695560714545947130529724467545
3023 47 511953545751.66177107863339231208 6789457020754215256411685440174304374421
3079 41 0.00000000023207164115 2141735462113012348434191
3119 69 0.00602643768003535676 1049324059291363104640660260593606354947228
3167 53 260425057231369.09972689242486572823 179940200099564565183145128420460525695159379
3191 69 14.95753417523273502334 628315606611285122323089495662377335179881369
3271 27 0.00000001241846857681 243743805808037264960
3319 41 0.00000000000000931719 33642982599580090033407
3343 19 1.58620548101554534230 202212319807498314139
3359 69 1.67365772978370467795 4049500549734207136741889632539175584626802773
3391 37 0.00000000047269143915 3101068731835541701552139
3407 57 28819363.52788452185843622630 14560069825940266325266106631738472810251876
3463 19 13.77281385527310381528 8436425008239920177883
3511 41 0.00000000001434615685 11283089506997160477570507
3527 65 7976156006.93765520950738207851 177909675370561108532941247559470051694366241471
3559 45 0.00000000000454262476 2360984303797804985269627200
3583 29 0.03920861329093863272 5633558142934297942544477
3607 19 36.90684229826591202567 6563789114878102952373
3623 45 26.02198199101404503659 1367770373787759276795854320370634923
3631 43 0.00000000000255623850 427172018608825201653180173
3671 81 68467807.61967835918310312766 117843219225119087861773569889261690958967498129448976645
3719 67 0.00000000387780125945 2100976923197097618349171255793507783224063
3727 31 0.00004718946253999069 2444912707864422958673293
3767 39 0.14327587591455739687 19175435293484578149783919131747899
3823 29 0.00002507286357856734 112417612198200717861969
3847 23 0.00001533345082846095 862130992415445336857
3863 61 15711737.48318757316819605069 538394367773173591693084396367727593976141647921
3911 83 0.00000000001135189188 14499327197980399167915624859957367185588546363629
3919 39 0.00000000044261630153 3669196191866220345463135468
3943 27 0.00000000002095427812 24733668567484868147
3967 33 159215.22350765659455009473 1699553443719169549240379327572
4007 57 0.91990088235951608333 24479483802292931184510605398988563448578689
4079 85 0.00000127446973332763 321963672249515003702405195900159942517697859377925616
4111 39 0.00000000000002095214 33609809857361575194185877
4127 49 264235.35416434781417718265 8507300729794243618442185373646968515974841
4159 31 0.00000000108677317318 8139478924692126488523677
4231 51 0.00000000799222631822 4299570633402618922705497439694423
4271 65 0.00000290128128404136 28283186967833318171568219977944584501489867447
4327 19 0.00017450275228592056 1066852592208273415311
4391 79 0.00001251482541320985 111218267845699128837861765031864433022917071402312905
4423 33 1.13324073727367674386 572395522360267105755996447652
4447 17 0.00000077492121857457 39872685366747226231
4463 55 115564.52263908769265426894 72780887497608197802303423424278364999654385253
4519 29 0.00000000000000004313 3597009679993991314033
4567 33 0.00000017729987000499 1945455740103468168767260495
4583 61 551441731499.78356036276104299918 1438973413788257170719455133810181008462888610212680883
4591 49 0.00000000000000079237 16108217968515978652127771698061
4639 51 0.00000000000000004301 101877348949955205680678906825472
4663 33 0.00001145384710376496 1169504442816257396334162500
4

2 The case
q≡3
mod
8
In this case the curve
E
dened by the equation (1) is also dened over
H
. Here the prime
2
is inert in
K
, and the curve
E
will always have good
reduction outside the set of the primes of
H
lying above
2
(assuming
q > 3
).
The Deuring Grossencharacter
ψE/H
of
E/H
is then equal to
ρ◦NH/K
, where
ρ
is the Grossencharacter of
K
with conductor
4OK
dened by
ρ(a) = α, ah=αOK, α ≡1
mod
4OK.
The algorithm computes the values
L(ρχ, 1)
, where
χ
runs over the characters
of the ideal class group of
K
. Again, thanks to the above formula and
Deuring's theory it follows that
L(E/H, 1)
will be given by the product of
all the
L(ρχ, 1)
's and their complex conjugates.
Our numerical calculations (given in the table below) lead to the following
conjecture (see [5], Conjecture 1.5).
Conjecture 2
For all primes
q
with
q≡3
mod
8
, we have
L(E/H, 1) 6= 0
.
As it is remarked in ([5], p. 2), in contrast to the proof of Theorem 1.3
there, the authors see no way at present for attacking such a conjecture using
Iwasawa theory.
In this case, we propose the following conjectural formula for the order
of
X(E/H )
. Now, the Tamagawa factor at each prime of bad reduction is
1
, and the following conjecture agrees with the Birch and Swinnerton-Dyer
conjecture for
E/H
.
Conjecture 3
#(X(E/H )) = L(E /H, 1)22h/(Ω(q)2√q)
.
Below we use Conjecture 3 to calculate
#(X(E/H ))
(i.e. the analytic
order of
X(E/H )
) for all primes
q
congruent to
3
modulo
8
up to
11131
.
q h L(E/H, 1) q#(X(E /H))
11 1 1.73845792121760807790 1
19 1 1.00717576250064706853 1
43 1 1.27416027648354776885 2
59 3 3.27291981598555587930 5
67 1 1.22243364144817892444 3
83 3 0.03764760689032642372 1
107 3 1.03936693115122887083 9
131 5 0.00697158270425921776 2
139 3 0.06083483360363034315 3
163 1 1.23339224060989144293 10
179 5 0.06195226194655516123 17
211 3 0.00822152771334987023 3
227 5 101.51957725718817748183 1524
251 7 0.32436912039697336340 343
283 3 0.66685505848109321080 53
307 3 0.78609012909015461469 73
331 3 0.14907707956876846359 49
Continued on next page
5

q h L(E/H, 1) q#(X(E /H))
347 5 0.08080910761305406985 250
379 3 0.33390038219003027566 117
419 9 1.67741166405082133222 40225
443 5 0.00285060561331757659 156
467 7 0.35140804797178265726 8190
491 9 0.01884246766690787210 11387
499 3 0.46414182882437918271 395
523 5 0.02170338307342622816 242
547 3 0.41073019842431356261 316
563 9 2217.38322210227025102910 7489893
571 5 0.29829944260019244950 1956
587 7 0.25898213029907196441 31143
619 5 0.00509394359043077170 479
643 3 0.60354061296945788027 893
659 11 1.79783553470780040531 4006249
683 5 0.00159958302432158035 1447
691 5 0.10350577224179846609 4018
739 5 0.10706905313737192318 4561
787 5 0.01395924800465447269 1425
811 7 0.17939743855163310334 60156
827 7 154.81828584920445528361 7220288
859 7 0.00003489503731419296 1202
883 3 1.46744585216089836663 4031
907 3 0.59650901915093688786 2888
947 5 0.03074953029629406725 62879
971 15 0.04674641781017625752 846868715
1019 13 0.23524712311785247652 481254336
1051 5 0.15568884564868710538 48000
1091 17 0.00217725207751161362 4271088999
1123 5 0.07364280939031303020 23322
1163 7 0.37473402025732946137 6435488
1171 7 0.00082674907382398778 33427
1187 9 8.57806631683350786688 196575884
1259 15 0.04793034848358401730 23807653018
1283 11 8.90099754034007899869 2309889447
1291 9 0.00241815361075660441 1270411
1307 11 16150.89301909056086676059 186585360146
1427 15 1.56491207854150961696 208776957205
1451 13 5.78796172237250084560 198287772553
1459 11 0.02184103208570492546 81647642
1483 7 3.19082650762232731055 6018956
1499 13 0.00107052328295353035 3762443612
1523 7 13.14472327791761386093 432126977
1531 11 0.00934308869252333838 91517708
1571 17 77374.81778094942048006157 4227257131159937
1579 9 0.04967065171792227175 18842731
1619 15 0.01588367050528775720 287671778919
1627 7 1.14234502669777570952 8445329
1667 13 3015.72287790923136695989 5415661616355
1699 11 0.00000951105549164743 7161466
1723 5 0.59636225379566621680 627691
1747 5 0.63250816243545283913 545202
1787 7 0.85334738271374115107 618910949
1811 23 0.06992899432395535990 16061273092160342
1867 5 2.38591921224867532372 6297927
1907 13 6.99689763033106016118 3165747533075
1931 21 90.38028537368310596632 258729934559477369
1979 23 4142.79431454502235459380 21986676642147049263
1987 7 0.00236988066983797578 2280430
2003 9 0.01742633984133732059 2517289989
2011 7 0.00053306208627154529 2638203
2027 11 0.14670352596641644330 58540694152
2083 7 0.05598543472754948453 4585113
2099 19 0.03659270303763838186 2787484250243039
2131 13 0.08176766229504099310 89178286597
2179 7 0.00000665857111220152 527657
2203 5 1.91715188964021586919 15965226
2243 15 455169.94065435589748159377 81753149480991824
2251 7 0.02300309205779565400 64040332
2267 11 7.28105312585640396026 4870756173147
2339 19 0.97878697785561566356 59954610410365278
2347 5 1.60476528633349239697 7286700
2371 13 0.00000800495831506833 3115285126
2411 23 504.85388397709402086354 362000857310467738783
2459 19 0.93923549011641697155 117489182977955129
2467 7 0.09990402861216668317 100745543
2531 17 0.00014747927341115960 306040338160927
2539 11 0.00011234010687514967 3085577520
2579 21 15539.25782562482093805923 758227028237459575815
2659 13 0.14349748145910982056 1922642144666
2683 5 0.35221319600572141285 14543262
2699 15 0.00003740841184509724 50729716969650
Continued on next page
6

q h L(E/H, 1) q#(X(E /H))
2707 7 0.17461587946412744900 348539119
2731 11 0.00000353700779516742 321349458
2803 9 0.78284831007160905719 6224866339
2819 21 0.27671024228777928761 20477277172589698869
2843 15 10.14431840386199781185 4022140478178599
2851 11 0.00056503632880051287 11870467976
2939 29 0.00624713894355604616 105268137875003312953547
2963 13 1.32077115136644560933 319550765817053
2971 11 0.03190494829015117475 324491996326
3011 21 0.26180404016133656775 33950238559165464896
3019 7 0.04312030585447679517 1802300809
3067 7 0.50159638459199363857 383973387
3083 13 66.44365839794315801657 15879087340845386
3163 9 0.00000102202290136419 76089107
3187 7 0.21740590516706865825 569775265
3203 11 7.58674726638825494993 378391105040096
3251 31 1.49283692874036018378 208144942072228395025506250
3259 9 0.01121435077834574673 21243653932
3299 27 0.00000054505993369606 403911133854039617472
3307 9 5.24935939959546086799 190357002279
3323 17 7353.12213893642943109960 56954453746238785334
3331 15 0.00013628808312750543 12824146340774
3347 11 0.00693687813140206147 15997504388590
3371 21 0.00000905587936280291 1798520024157799359
3467 19 0.14118916689423367305 2179702892746704617
3491 23 1521.81323914654475213527 2216917282043791853812677
3499 11 0.00000048245603947392 12562962745
3539 23 0.00000000024192865064 502868601762633637
3547 9 0.51723990155998270507 64134211184
3571 15 0.00001877157784780692 5298557564473
3643 9 0.19035162141447944342 196656106527
3659 29 0.52058334069619626047 224345918498470661616572059
3691 13 0.00010142020688338228 2190791538023
3739 11 0.00021638831563001181 281259932931
3779 31 8.11514180382761610358 29319627255957776379402310856
3803 15 617.93079127720341400907 6125701430298575603
3851 25 0.60046598691752433358 1242978436326110671361250
3907 7 0.03397552970283538243 1708500805
3923 23 0.06601853781292555224 16160511049152622462466
3931 11 0.00034265756041784101 281879748512
3947 17 1.75074743805447665390 28850358440768828838
4003 13 0.00530457688125282635 13618654340552
4019 19 14.89793441535627886639 10201728430786087533096
4027 9 0.06278002336060817839 27507874104
4051 11 0.00000004836198044776 32711170970
4091 33 0.00015048845055910840 13119103199363816872272275835
4099 15 0.00009653216942583305 135256029258383
4139 19 0.00018105359839238306 80074451508066303468
4211 23 0.00001204796805032161 4490981695211565226634
4219 15 0.03461154251357343869 12962166563665575
4243 9 0.00066320718752067075 4758420837
4259 35 0.32159852619627858305 16759504772930116391482641893012
4283 21 8773.29312623157893512197 1535643789741168641279699
4339 17 0.00136435844805308578 41415425722868069
4363 9 0.00169672983900952063 12072493829
4451 29 0.52221992024973798611 57243468904104560776219171927
4483 9 345.39066341572129604016 90594797052049
4507 13 50.23492940730687592262 8000361069297882
4523 21 0.47925191687294653824 12696689618269479222463
4547 17 0.00237327326500158548 3365011327793496632
4603 7 0.67370851258063680284 37507414772
4643 13 0.05750243316638869235 143101155917879067
4651 17 0.00001544101918254929 15235214493740502
4691 21 0.00000333535197828509 2621016573927853606584
4723 9 0.95386187065080598582 5224661311515
4787 25 196299950.09431612703528249287 1407786352476914766515861602276
4931 35 0.00000002277116284325 448007986303112905715567652356
4987 9 0.08774530421699530811 1780109232599
5003 15 10706.38070422879767696786 34395116945398635220139
5011 21 0.00050859723642713633 91159910603652230720
5051 29 8.49904519468908134660 1141170772780262770088006751376
5059 19 0.00156551345932179882 9867629103745428509
5099 39 0.20398131340164447660 1184616873402209140039591463547587231
5107 7 0.41759384988925603185 40655661145
5147 19 9.89685243132721839321 110480338090747788980762
5171 35 0.36860175091251048590 7458685927940450075817149649785164
5179 11 0.00241753460843546585 432857713333757
5227 15 0.02590550552009961454 28726219588915216
5323 15 0.63466138387854294475 101661479910418479
5347 13 48.38774889967386398841 60377540719495011
5387 23 46969768.16794671957479225493 59861891852089030100409766615
5419 13 0.00155877501356795214 3346444025253455
Continued on next page
7

q h L(E/H, 1) q#(X(E /H))
5443 9 0.00772315292672455124 259851370654
5483 17 0.00000850543167234778 12838230594323889723
5507 23 1196283.61113816912742549060 308977010562437453037385949809
5531 23 0.00787997073850461781 114928716348176366112944426
5563 15 0.17610681503569755604 289650044515887808
5651 31 0.00001430611962530720 1053842731936249031364945295419
5659 19 0.00000095193065835809 2612065297147014949
5683 11 44.34550258638220009604 12037895727172693
5779 13 0.00020942032710240135 1281689314164362
5827 15 0.00301443397675326391 76300991821550264
5843 25 5443491913.33884952923429281132 965194290530223430718350605440606
5851 21 0.00045098763594953089 4466493231539670036837
5867 21 2.55857313302494265980 6273825474355118152363875
5923 7 0.58161114692979764321 287690477472
5939 35 603122.99403995879567858207 1661965438061076810127871901025304745853
5987 15 110.46116247014125681999 13865640754944135873431
6011 27 4.86492240549697162717 87216830057513280930236553759140
6043 9 0.00721258251692584701 635377400757
6067 15 0.00083214765715847858 66156492166979308
6091 15 0.00004657801554426414 37563167157316048
6131 31 0.47402035862479912750 10588808624066473635049652220685900
6163 11 0.00109962773126547504 212505143679808
6203 17 16956.91608309776740115902 17758388633281304578194836
6211 15 0.00006166346655870246 143325584038005751
6299 43 2.05775857985552601858 1388856212566707742241274725107204020014609
6323 21 0.15649673018585925176 5097012104044281804792939
6379 17 0.00000844841271024925 2826990910532861162
6427 9 0.04949970413006985692 3642618058167
6451 17 0.00000000015961944529 4513598851205200
6491 31 0.00000661570599992748 26061044582689107840212347433528
6547 11 0.00049212375667197424 28027440443663
6563 23 331287.05252863171559172025 875165582628365469003240470630
6571 15 0.00113860246370370679 6769437817419744845
6619 13 0.01464489531673771186 228222213173218843
6659 23 0.00004608564886020156 3960701905118801624128927055
6691 21 0.00076792513778405932 44437713772698490604378
6763 9 0.02842453885205402194 12452915791145
6779 39 4.86957703804081099214 69700200603218027480784056371040211086176
6803 19 3887.42979178406639019887 18785753293448529197999195649
6827 17 14.01183337740201353984 3988442850286746357347994
6883 9 0.00267729911739092469 8323175699073
6899 35 0.00001267331994634709 216639728383074935163216743914068125
6907 17 0.00034176219626136351 7191713364116285807
6947 29 9807687.61625312683632853546 1396721892374430002050625825823542808
6971 45 144.95259877715406861536 16300433495176531353395464824906912329257404117
7019 43 0.00006851095988046843 419823130126477043502027992927631092428380
7027 11 0.00126334189165518431 372846808979421
7043 23 79.57658531732701207353 42967244325677025460819829585
7187 25 1941.80715024988578872060 111790555942583682538467190725164
7211 35 0.00000013542177935543 92297726998496877729409205982930880
7219 15 0.00005574430738945572 1293006824668014040
7243 13 12.33651259315172930941 1756240279022402156
7283 25 2.49002172917008268374 10685692353679063484178785461948
7307 25 62171.26156087145465015128 1419081250825229997211797415503825
7331 33 0.00000000032502361559 219455139795728684736426909205613
7411 25 0.00015095236624874787 283024230782339795658408815
7451 35 93950.78490910382748157143 1553319522388195483106386505386846710579688
7459 15 0.00003036649237987241 2337968558206904465
7499 33 0.00000135124978504196 60792334311439101075808649223338741
7507 11 0.00010611225020431393 506903678439547
7523 35 33522.02988333545407545411 15470691108260141377130066889443447728251
7547 15 0.60032452144273867428 258920923898858718493581
7603 11 0.00653892833763582491 1434870589808453
7643 29 179270.99125915725948310242 2288294534095276709215678772272550333
7691 43 1936270.78489178696965784529 3624801748457692072742156116148403862768295124739
7699 27 0.00000699702201774193 2304258410337272194320291613
7723 9 0.01895970678708005242 46425972727345
7867 11 1.61910857940903382775 969216189901440388
7883 17 0.23552172495493403317 28080473969786372085402707
7907 21 0.00175081994226267973 568887022906905104307160000
7963 13 0.00002874140087673655 1773163400695878
8011 25 0.00000000000000681929 5972318120577118158596
8059 21 0.00000000008523807585 384252254712642261955
8123 21 2383.30719868549773975742 18940429636369330011648211950850
8147 37 12176247.61615672604210432982 108543217201536511344104411121816314211436357
8171 21 0.00001153050501312074 43709209245428902074487555857
8179 25 0.00000000000408693701 245379074777785127542855
8219 35 0.00003219371470307854 320575044136527674847638735156405728284
8243 21 0.00003184119831052584 169936812859198942971657899
8291 47 13.82276752176918097863 144216236301930044357917532496691060210143449492929
8363 35 232.06334304127124142727 15312049368913584760165020656408258883985
8387 21 1545.33342862710955449647 646162618485014801463564772032
Continued on next page
8

q h L(E/H, 1) q#(X(E /H))
8419 19 0.00000012646740535682 7680287320591089475912
8443 11 1.17893537822797473748 35478826850611232
8467 15 0.00021223474007323571 1265829019458818985
8539 17 0.00000478356805956157 2275177608844149620235
8563 9 2.03683466086338426663 5420212257148597
8627 21 3853.12162724028605983782 198171648305412280420765428378075
8699 35 0.00002481823232900755 11240439275853225767486298164708391703500
8707 15 0.05248611135357774651 544957212659766295149
8731 17 0.00003500145787893966 1576791412901313685993
8747 21 484018.07264403016140460258 288450573402627146790548293522904
8779 15 0.00000030582300865004 3039062315648643705
8803 9 0.01552030193111727803 126149700575776
8819 49 4034780.63025405890013810127 23921217585664253728980536953688630289914216264814750750
8867 27 2.17002992458273671577 128281233388318287503394675078625248
8923 19 159.33124314583988814840 14174087757188097620330473
8963 29 1985418.83735640580162530103 1118129255137682166151656920634902507171
8971 19 0.00000000983100532208 2258409802656205931201
9011 33 0.00335393090412302635 3045909543390800207752459587622105801155
9043 15 0.10664966467865754486 247161131467830177656
9059 39 13.77046293947696022352 1591196881664881653149442733268746342271235079
9067 9 0.00731970822333483793 111048528074239
9091 21 0.00000000236926748020 236934738121647151313216
9187 21 32.52092107710830419611 366682928077533301738876375
9203 31 1023606763.23245034828055208642 10450577199144964797897931682954499342069972
9227 25 0.03059726342701622948 61065081813711319346093946656236
9283 11 0.00312668644337035262 114634424721388682
9323 29 158279113.16389204676979182791 198428451440095157640040934380936648153653
9371 49 0.00000000381610952189 9868042349887290226589618012608800982902276255554
9403 11 0.00062444568609082732 9557020329789716
9419 35 0.00004080832458224277 7755628001204338180900900374435880983024
9467 41 85262.97994248194021607026 712735551990570229824239164893859490851236314718
9491 45 1.93118062921922617356 699904157096559596488802074953836695863315457589115
9539 55 0.00000011262029208767 1854017230259878781685683081695866642830423627047547360
9547 13 4.55503594583652426666 636512933633007687775
9587 23 100123.63380876703245300853 36563316420884623249966367262977278
9619 19 0.00000012220071093949 5176494409557611828238
9643 11 0.00824565865962835292 516445608289196035
9739 13 0.00000131331766917028 3753135247337182744
9787 11 0.26005393619869399226 362054016521593344
9803 37 5930542264.64370531353643671288 3056997114849536281991895391097536748139790872620
9811 21 0.00000000057298287826 127050303874152535296768
9851 45 0.00000000000000654205 18104457473462858976938437755791606152672401
9859 21 0.00000242965527955730 28849640043711849588277760
9883 17 3.13476671971555114510 242746734273754376709691
9907 15 0.00104023793825974741 106327569146508318840
9923 25 0.00000242632276300466 2387019558186614148757292853595
9931 23 0.00000000000063190232 807612994249556860279774
10067 21 518.08680600795693371589 342288622319158792879184584048129
10091 39 0.00000478306284603539 33197255379943602421429579685905522266248888
10099 25 0.00000000001780723538 52822827121201782525001225
10139 55 73803.78400959271904263740 33150521228939817885342694776696740143379123299513891375082064
10163 39 67991.83407099603855149887 440513245214192270403084087547571959843972111425
10211 43 36.21163226407725524787 2018436122269325072524514224058369879912299468812420
10243 15 0.08816299542047442521 1839984519589335519209
10259 43 652.29068417959804070259 718692371932632451954249149920382743582780046485373
10267 15 0.00014937068406596087 382939450567034881777
10331 55 0.00062202944015487748 6664989351016927856864678973892159602391574133103405618788
10427 31 43.44997691536163284782 2845156404392349071355292880344541653644
10459 15 0.00028151807209305816 4605303213234756184743
10499 41 507989.87585259730539676742 36955062681702947421893940479768814784946674217168545
10531 27 0.00000000000146797238 3351769586174993494323439275
10627 9 0.00017318804748393473 398694874516979
10651 15 0.00011268583573677928 12553194111400362264301
10667 39 25506036465736.67223882308179488976 44811079267440631723932527212656541784290245322216891
10691 45 3520.56248376976938840747 232694011311673350014283799021469164888464834922982077
10723 15 0.00000092340639534283 21063173152441592413
10739 37 0.00000000004343365251 63654774159741723864112646151198426204132
10771 21 0.00006625182262625102 6376645704546135373491729472
10859 45 0.00001929088385542839 144220156391029884341565008365356176802926959516032
10867 23 0.00238767061834684639 10525038031112725310622398635
10883 23 515416.17141491515842049134 87346650878073993681932895672040241977
10891 19 0.00000016242472588156 11884817420032518255654205
10939 27 0.00000000901518555777 384273411889144154980660519291
10979 35 0.00368603736420883092 3439622056704005260743458738409400370601464092
10987 11 0.00349823665853601300 20844399361215150
11003 27 2285808.76686708228105121178 215352519101955536146438795127413473169151
11027 27 1.79342141479218417410 1218295757801391198254967804816697855
11059 25 0.00000017029649146888 339791808371879703974813638561
11083 15 0.00029395864104112152 69130942815185383643
11131 25 0.00000000011466927704 779081246955355170396973552
9

After all the calculations were nished, John Coates informed us that for
q
congruent to
3
modulo
8
, one should actually take the square root of the
j(OK)−123
with negative imaginary part to get the appropriate quadratic
twist of the Gross curve
A
(see the formula (2.2) of the paper by Buhler
and Gross [2]). Hence, we should work with the conjugate of the equation
(1) under the Galois group of
H
over
Q
. We have checked for a few small
q
congruent to
3
modulo
8
, that the
L
-values, torsion parts and Tamagawa
numbers of the two conjugate curves are the same. As a consequence, the
analytic orders of Tate-Shafarevich groups of these curves are the same. John
Coates expects that Tate-Shafarevich groups of these curves are actually
isomorphic, but all is not totally clear theoretically.
Acknowledgements.
We primary thank John Coates for suggesting
the problem to us, and for very inspiring correspondence. We would like to
thank Mark Watkins [8], who sent us a new algorithm allowing to make great
progress in our calculations of the values
L(E/H, 1)
.
References
[1] W. Bosma, J. Cannon, C. Playoust,
The Magma algebra system. I. The
user language
, Computational algebra and number theory (London,
1993), J. Symbolic Comput.
24
(1997), 235-265
[2] J. Buhler, B. Gross,
Arithmetic on elliptic curves with complex multi-
plication. II
, Invent. math.
79
(1985), 11-29
[3] J. Choi, J. Coates,
Iwasawa theory of quadratic twists of
X0(49)
, Acta
Mathematica Sinica (English Series)
34
(2017), 19-28
[4] J. Coates, Y. Kezuka, Y. Li, Y. Tian,
On the Birch-Swinnerton-Dyer
Conjecture for certain elliptic curves with complex multiplication
(in
preparation)
[5] J. Coates, Y. Li,
Non-vanishing theorems for central
L
-values of
some elliptic curves with complex multiplication
, arXiv:1811.07595v2
[math.NT] 5 Dec 2018
[6] B. Gross,
Arithmetic on elliptic curves with complex multiplication
,
Lecture Notes in Mathematics
776
, Springer Verlag 1980
[7] M. Watkins,
Computing with Hecke Grossencharacters
, Publications
matematiques de Besancon
2011
(2011), 119-135
10

[8] M. Watkins, Letter to A. Dabrowski (September 26, 2017)
Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70-
451 Szczecin, Poland; E-mail addresses: andrzej.dabrowski@usz.edu.pl and
dabrowskiandrzej7@gmail.com; tjedrzejak@gmail.com; lucjansz@gmail.com
11