curve #3565

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y² + xy = x³ − x² − 175454373647018982254072256257832924x + 10629934898753117975527831597137904651351874164717568

a-invariants: [1, -1, 0, -175454373647018982254072256257832924, 10629934898753117975527831597137904651351874164717568]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 279371967069651688648184842896025457250762762386873734247421810 ☆ record for ℤ/2ℤ torsion, rank ≥ 17
discriminant (Δ): 296864597058843106036258594644492840156421488273232136489313771665616715662338189632055517498920442754074000 ☆ record for ℤ/2ℤ torsion, rank ≥ 17
Faltings height: 19.3150 ☆ record for ℤ/2ℤ torsion, rank ≥ 17
naive height: 255.0717 ☆ record for ℤ/2ℤ torsion, rank ≥ 17
primes of bad reduction: 2, 3, 5, 7, 11, 23, 31, 37, 41, 47, 67, 97, 157, 239, 241, 281, 4093, 4513, 8093, 45853, 314903, 743923, 190467271
regulator: 106079819419927210328610147.909682153623881954657505967976923385107779796
submitted by: Matej Jurasić
submitted at: 2026-09-28 15:25:24 UTC
last updated: 2026-09-28 15:25:24 UTC

Witness: 17 independent points log in to add more points →

(12569087689418726137/256, -189208271517191293529248388981/4096)
(-764864410043848396247/1849, -8862029914826854335957564416687/79507)
(-107519564050823249423/256, -414351938750941256859307458161/4096)
(4630461026449813268, -9923743962226000755677969504)
(-78538577331826056851696483948268224/943717478668195729, 143952136946727050346414924196239998012308433542714480/916775498818695152140426567)
(-35817560664922644974007157490715692/193141639381295929, 16280868859073546834968746716262317542887849821947072/84881661120148940616795283)
(-96803116930685123655955106471/369786826201, 44191456164940922095472619945745433799557791/224867738799654301)
(-279494461171007952269791/4734976, 1485282652297857900172614737886588175/10303307776)
(887355466244896598809785557708608/8651439816889, -26432778943465445235444834013002226756877798879792/25446765430929573037)
(168278825279734067656082752/87965641, -2131677825392197572529503981272973885584/825029746939)
(5951290915149067458293129807168/3405932997289, 14109894217816803025886446526801589140862591792/6285707247357803413)
(6435397289404987153787860288/12390583969, -336733079903519007364529605542703960775792/1379233073341297)
(4403778158435114517738953447847953/932074610830609, -291103025113009515653390064943115074610346408692973/28456141864973577217273)
(14684228521666819580323367089/25365340225, -1296968115190851958535259618749540106299093/4039810910934625)
(6253315660454021503905585913/1291798184329, 145203549378793337998953984680350092482332823/1468222937757364517)
(80201392352155662087539314850932/7139001515881, -717748685109885554117911859437348151953549623776/19074633899268800971)
(2097416341844778295011673851828226293762263159633/5455017105072595996527221711104, 41186379721186198071706093525659242983556159498123425699590665207052047/12740725974920217699650111299020209400400375808)

Commentary

2-isogenous to the specialization at T = -4834/987 of the rank-9 fibration of X400 (equations as in curve#802).

last edited by Matej Jurasić at 2026-09-28 15:25:24 UTC · history