curve #5181

y² + xy + y = x³ − x² − 14764248903496077732778111905x + 684876197616122984742933036421115128682097

a-invariants: [1, -1, 1, -14764248903496077732778111905, 684876197616122984742933036421115128682097]
c-invariants: [708683947367811731173349371425, -591733034740327069740130988315053191109160625]
rank (lower bound): ≥ 22
torsion subgroup: trivial
conductor (N): 854719095669670901420795176752574947323841009617302550504741228442740450
discriminant (Δ): 3342844948089679511007802042557695695165165440319624207715121718363785236399104000000
Faltings height: 15.0396
naive height: 206.1996
primes of bad reduction: 2, 5, 19, 23, 1747, 995387, 299079203, 3958651592872630315675970842186160379851222074109
regulator: 65397197186819225806614.18614656891493542329311510114723075794910681594182845426347
submitted by: hdeping
submitted at: 2026-10-21 08:08:27 UTC
last updated: 2026-10-21 08:08:27 UTC

Witness: 22 independent points log in to add more points →

(-11336790594577326729/85849, -14487438596926275979795431180/25153757)
(-78766284252841, -1165815190645163018780)
(-18782945860664729/225, 3900473103366240580344236/3375)
(-75146366115081, 1170472634271157603940)
(16990407331015, 662518129111099411348)
(-38374306093956009/529, -14262735619434171860919380/12167)
(-132365451636621, 565710988214161036260)
(-407492410499762945/3481, 184732835733824743466498124/205379)
(-127858967554121, 694543230979797436260)
(-6121556431955801/81, 852971195162324598713540/729)
(-94077767242841, -1114098965377526578380)
(-112081901131881, -965229929890208767260)
(171987950472892531/2809, 15203198531683742099261940/148877)
(-627997690565291177/6561, -587268526220181428563045340/531441)
(207981690583344871/3249, -8432862992676483445821020/185193)
(-1009403110104689/9, 26037449749020106135820/27)
(-87956262563337, 1141502542403944163940)
(64307310558455, 36951988345069243556)
(-66151881393211/16, 55272002336978391748415/64)
(83369798777119, 182886454170385466140)
(57745876752256639/441, 9245984600015853065877460/9261)
(134362303648119, 1061504097616933512740)

Commentary

fiber 133 of the x1152 K3 fibration, E(u): y^2 = x^3 + A(u)*x + B(u), specialized at T = u = -27/98. A(u) = (-98446928369392297987858107)+(753656616847155064495894056)*u^1+(-1304662621155467165949437316)*u^2+(-4515151930070151665108601288)*u^3+(-4977701063274588505383095730)*u^4+(-3216158287288357141650052008)*u^5+(-379643944497665001753404196)*u^6+(387597384378623366437498056)*u^7+(-52947472367727871375145067)*u^8 ; B(u) = (455529334666944868668501131133770300694)+(-4406091141778868895239142866363000327928)*u^1+(8711059880694673976260419900526416998364)*u^2+(-3793179221642104157592128359195176701880)*u^3+(24216919834250864412914656434287671696410)*u^4+(200113292396607048670446102942302942345616)*u^5+(266455804027297683905238442306815065604168)*u^6+(107441306099609449962047233442503188641616)*u^7+(-2582417909382033421927144160479025688150)*u^8+(428474205328592695582331592965376042600)*u^9+(2199423551161536880161124686788281724316)*u^10+(-1449946611991997592459259221798535014552)*u^11+(194470684143269906473287684559611258726)*u^12. Rank 22 (exactly): 22 independent witness points (17 from mod-p Shioda fake 2-descent section lift, 5 more from kappa(z*)-covers of the 20 shallowest 2*Lambda_17 coset classes, ratpoints H=1e6, exact halving), 22x22 canonical height det = 6.54e22 != 0; Bober analytic rank bound 22 (kappa=2.8, GRH), root number +1.

last edited by hdeping at 2026-10-21 08:08:27 UTC · history