curve #545

y² + xy + y = x³ − x² − 26265180538584774243042039506046399736527297073029718577x + 52202360216817077639513658746328510627841035595380523866032931014069689213302319201

a-invariants

[1, -1, 1, -26265180538584774243042039506046399736527297073029718577, 52202360216817077639513658746328510627841035595380523866032931014069689213302319201]

rank (lower bound)

≥ 28

torsion subgroup

trivial

conductor (N)

119991230426783777973113135447507300592589917066785825793402245650519713138188903012278278050834538793484000598995158469282650892654548362210

naive height

394.4522

Faltings height

30.7237

discriminant (Δ)

-17602783132765139090384259836796019052112608754947579945005541162913554998303656371939246666148928267750952176204939066187058828627853309348153809090202913107968000000

primes of bad reduction

2, 3, 5, 11, 13, 53, 103, 149, 323129703998262103, 63396844598452836179252569, 43202157024604289771645360996569357, 12951546843027284240909042888616045358018757136877687

regulator

50676608533767495206379711781029227.45685838948902212788102895896180652855589127265937926302488

submitted by

Mundoamundo

submitted at

2026-09-02 23:19:13 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-11 20:23:22 UTC

Witness: 28 independent points log in to add more points →

(10249149541458104823715958705, 927161900900819012498784941492090165014072)
(206086135982829487797801680479/25, 78647264948423100947024414527061350127053732/125)
(-23681730404516430512257536881/4, -108345134929851189443797382365453779760253/8)
(2540529445604543531499615079, 43269031816537721367292702784653943224964)
(945080857836338207968722806191/361, -255165995791242029377656827558369771229915364/6859)
(-352805909240496678446238271, 247840587187769154573350591073078248988664)
(4306325178462040162682268151, -137674710643205973996782041283479959859276)
(123592833204061043780407404259/25, -25974952252231911852477340726930219994384912/125)
(4438478535356742919869839071, 151866238391574047256112222040902871572844)
(5926381233093123821394304351, -323560220640653373856932501007779447078676)
(-5691726229749521608965297545, -131563108753537738040086971926711424704428)
(2272350936059309131355449751, 65208248716262248972749175908322288197524)
(2428879072726962764456386711, 52311300243877216586296014567651332832404)
(6304488690338559864153143101, 370399346401019765188593619086469361616824)
(-1967051372669651532987579281, -310251873924701621627591247278884861929316)
(2653712265613083282370597751, 34497409008165119802879293482279373501524)
(-20780467858803378599466604481/9, -8561072239815255669311411364148428613254772/27)
(191970770833960111322487540871/81, -41660584854017911968855727739931278355380044/729)
(2723825597417367505197049964954986139/1254363889, 3266818214629435322294489339928215935120288327275680212/44425805856713)
(-4510104087455073096343891049, -280928559828490580529986744556459745120876)
(3426763494016361279180198101, -49368832569287258274994171883405444234926)
(447518175472445778469465153148182111/95667961, 166598822228775703554423928267226788798935586435391684/935728326541)
(42821407462815374794517520195839/12769, 61821745281207351739447387243606748858419768108/1442897)
(3083999638216344658350952471, 23078980031664735787195217118615386859284)
(-1820896959483139705621876649, -306579617620831896708476765948390066757676)
(4503573198540457847961374551, 158925898801829922452288441207095901747924)
(22846584386230514901471198855104480791/405176641, -108766308497896022949238252340039224331490748360578491564/8155800606689)
(31290413189138627922314856680876221/15649936, -5425841195218989097616854954538054049674596317223181/61911146816)

Commentary

Fiber t = -13219/2030 of a rank-17 elliptic fibration of Elkies' K3 surface (arXiv:2608.25406): the 2-neighbor fibration that also carries wgxli's #391 and #390. Found by a Mestre-Nagao sieve over t = a/b and an iterated 'bootstrap' 2-covering point search (fake 2-descent over random cosets of the growing known lattice); 17 points are specializations of the generic sections, 11 were found on the coverings; the basis shown is LLL-reduced. Search run by Claude (Anthropic) with Seth Lupo, 2026-09-03.

last edited by Mundoamundo at 2026-09-02 23:19:13 UTC · history