curve #4238

← home · all curves · JSON ↓

y² = x³ + x² − 164850105865145659274292935107078920908x + 818175125669547987173702806941070455529354454402883846288

a-invariants

[0, 1, 0, -164850105865145659274292935107078920908, 818175125669547987173702806941070455529354454402883846288]

c-invariants

[7912805081526991645166060885139788203600, -706903308578489460965556055686246823448358613914930372414400]

rank (lower bound)

≥ 25

torsion subgroup

trivial

conductor (N)

33272442660136729466590673810285498084436714991504530200336600126639243963197977438568499471400

discriminant (Δ)

-2472166709253325333479319934465044231752225241395354061996808564498810916685998009674189665929270134415462513120000

Faltings height

20.7975

naive height

275.6165

primes of bad reduction

2, 3, 5, 7, 11, 13, 17, 61, 107, 6467554945998766845309751, 77196337530513021426376013464805181856021978117810742058991

regulator

821134000305051465584043362.4649002253070434962662491182464563095141490348826194145662734

submitted by

apkorte

submitted at

2026-10-15 08:38:18 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-10-15 08:46:39 UTC

Witness: 25 independent points log in to add more points →

(111765820794798158518, 1174107043352296859540694269310)
(2859708908252356356286/529, 112170510933097612739582154674358/12167)
(10147040428942970698, -13791242554614047801538157590)
(-14255373933311705357, 16469955524671092971549890560)
(124393623936245709847/49, -6996191311115900337807976568670/343)
(116456331457778179054, -1249403892595379570003240389194)
(-571969361245061822, -30203927165955365527970312190)
(-6387676526320952333, 40131679153972837906872148752)
(-13461266253416728217, -24454207499005889925712782390)
(8573458847295610798, -5918169630028382063716969590)
(25722929569619838778, -116609670550174723791026390310)
(-12992551374620939702, -27690664219476941702286522090)
(31562183576387637703, -164488575395262177770158257810)
(49615037498091845344, 338783241242130989247146645316)
(104708424535034561713/16, 283526585982138536332358848785/64)
(4771206214914327718, -11842928882696438086445195310)
(45896786126162900458, 299890567426775646351030390390)
(9569970693843016558, -10817757839934886102733581110)
(429257447868412754794048/73441, 146684876136576960200672789434959660/19902511)
(-457631483108564035223/144, 62542416725265965776988553030205/1728)
(6833453112667827238, -3282550792906126414569052590)
(7731558138550529722, 2407536338732732017244477862)
(-1674479122407794250878507/306916, -6705300268320563068295788167416592615/170031464)
(81911276959978334398, 732731401703850210935118755910)
(1491648518565636544190009404/337273225, 82080704299633098121059627069869606596692/6194022777125)

Commentary

Specialization at T = 239/750 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -32332676430258071809200T^8 - 392497065814316713973280T^7 - 1605286506697565663751840T^6 - 2016536429413138685637600T^5 - 1643121380463324517697040T^4 + 604907641354957338809520T^3 + 2182672220879703769538760T^2 + 284764930480378808601480T - 903774960127741440937707, A6 = 2152872548702497670524683858518400T^12 + 37926485183963444897543405698016640T^11 + 273497207298763591399877065140889536T^10 + 1038319980551223501714889156644738240T^9 + 2025026435470961119972063737624451920T^8 + 768757757929830231396275004132348000T^7 - 1622968189285827532493125262292017760T^6 - 620910140516148584947737138229395360T^5 + 1602335569642802776124896462486427520T^4 + 351810003192753644435857685852379600T^3 - 1085870391509042811982855754241980040T^2 - 186927270560088531571177963463660760T + 323553892643757904650299161814986470. It is a 2-neighbour of the fibration given in curve#3900: its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-15 08:38:18 UTC · history