curve #1895

y² = x³ + x² − 16713576334398827023524947331020043757625x + 830828885443919870767137156222061509352603811437396554796875

a-invariants

[0, 1, 0, -16713576334398827023524947331020043757625, 830828885443919870767137156222061509352603811437396554796875]

rank (lower bound)

≥ 26

torsion subgroup

trivial

conductor (N)

361938467244585346401451369160427188614226141382317123766793726786123063944791757390100692461326216920

discriminant (Δ)

605686367000225385731131122380500908068123441380773503873383853690223633717962791389609145067423743729668741635892000000

Faltings height

21.8960

naive height

289.4647

primes of bad reduction

2, 3, 5, 11, 17, 41, 593, 617, 1981107112027, 542725489630303720896417580540198304251282216587173718763371775695662086752829

regulator

17750303554585349574031844911.4437145551289722244047825486527598879355825945102174849926866

submitted by

Ritik Jain

submitted at

2026-09-25 17:48:09 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-25 17:48:54 UTC

Witness: 26 independent points log in to add more points →

(126035432017289631870, -852285223560572822229911985195)
(288555203894921867336215/3721, -7438455577576507752781859909170710/226981)
(600102148971315479599, -14384412239719906539105081833790)
(-13623395142513977201845425/273529, -177510984031636854235339758827865456150/143055667)
(23833260481723747655425/16, 3665965017829690670641790059409175/64)
(102927367752307975335733/961, 15461625770215881476180535201179646/29791)
(7120162119533950063006645/97969, -104886039751306719131693318976009090/30664297)
(-88543477182180008285, 1271428665247625622927834259890)
(22790989783800160231375/289, -278862109890060480156476949606750/4913)
(8022310362278298030595, 718444358352888438342783030614670)
(63785272542213936430, 155766285217523576936340787005)
(762806817196677247438, -20783147889982624737897425976621)
(4613039991320921272033/16, 286062058247522822811459376151481/64)
(38085974742820552750, -499521095830823915382054775875)
(85658814938055028924630, 25070188438949429513172751168297455)
(1425236937406062247225/16, 14143300455512675233948123060725/64)
(901697825536140682987226055377627190616448992525/7026863512052181413761111849, -526558039725075765743989908280006929573688239683734217252791842211016550/589036595018979496243491533708519395825493)
(349164071222088101978875/4489, 11245000083160640003451083286728250/300763)
(-581436392210398747715/4, 3477636133306940215315853314155/8)
(342669110723166775561645970701/4356660025, 15293786744205776378980152126550679702422374/287561344950125)
(30008215158301523303313504265690361698130748635857/381246140912384089908710581264, 12751165056753368902901905963799531220435953173121371521194130320991834177/235400935526845164792052523168419412009854912)
(17649688959805674188794679593726317135/182951159004777409, 26685228339356660800992788667010271344885328031147117630/78253315592533008133343327)
(455930394852324494420821186189/10137469225, 420979578468108015770392005535567426259972838/1020691088919125)
(39131146267147592189168220728656858295686/41959114130048157289, 7670030064662006209884298557860939770359803085263459326740315/271793749994949998167764383413)
(-26994304230733247822847677926121052636038574185/201175989571055487390418129, 2313812661969021081728203706694543032257116281130070407026563317516030/2853410265935465630689131348714412660633)
(2829247047809171111992319893797305748431690518024724110/36612131245075798703217640315115361, 189362062562775642756622237130156324061758171120272537283792356266840510358609085/7005473842124805861343362054854182853136599072190609)

Commentary

Specialization at T = 99/136 of the rank-17 fibration of X1092 given by: y^2 - (Lx+B)y = x(x+D)(x+E), where B = pq(L+p+q) - pE - qD, L = 6858T^2 - 2232366T - 123553, p = 143664T^2 - 618920T - 6487056, q = 6170T^2 - 7098T + 7564, D = -806237487T^4 + 76922980194T^3 - 1226802677561T^2 - 689131950466T + 1104882072456, E = 815119825T^4 - 9557405330T^3 - 16096328659T^2 + 9197377220T - 46146438924.

last edited by wgxli at 2026-09-25 18:49:08 UTC · history