curve #1928

y² + xy = x³ − 1908533148950105622905349850110x − 1011676486474747706495075288533268546148531900

a-invariants: [1, 0, 0, -1908533148950105622905349850110, -1011676486474747706495075288533268546148531900]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 74554275095325030485358128944757956425405342640170
discriminant (Δ): 2769715403821110065464819901764196639923207041938771381658551546118740676549377011865856000
Faltings height: 16.2155
naive height: 220.7853
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 23, 47, 73, 79, 97, 127, 149, 193, 241, 281, 331, 1279, 95617, 314467, 707563
regulator: 35232768111818.30408522549427735448045047337740517428632582192753
submitted by: Alvaro Lozano-Robledo
submitted at: 2026-09-25 19:16:47 UTC
last updated: 2026-09-25 19:16:47 UTC

Witness: 13 independent points log in to add more points →

(1702856568936980, -26003335170777276894310)
(5273743246761194050009200573/2256328443664, 289503837623185793020957246580606680565607/3389249005855243712)
(593577648511773882843484715958054747648569028449306051737627412739685625148135996/67752979482474285003522925375563991994642784566010574862429431681, -14269646969113562915275106563273094853015169542998705922261683095450554754767462482658518416085234023213988309465806672722/17635695739370895529667828884239640154979943226793429063820277177487309698412863777128759752486079)
(33714212446183829375743549883803112350/17242315327109348661249, 118345818069053648832853318889606789235062149243057209430/2264088374858594565025793079944193)
(-154357506685143267458043245231303267989520132923843329540/186818890903789331012148463944392742330241, -87654627884884550244936452732074449594114399545859104608812317968450174083436800750/80747890270808383821858676261962449150894267145250105080112961)
(-2564604935496876237338085347908296315744159705694877140/3206642600026867735951049548777405275649, -322403631535786275185620300462421646469121447420043584377088020891553422003596770/181583271689937012812406095381626528080136075778906324644607)
(-2784500314909542093220/3644281, -4971803720008310700926558599990/6956932429)
(-1547795895574860132851344286914643098280469933513273656443621390147186017434711027430455229700371085101835541466914051119756399395841802880087862157222345224929258971412900/1862575294554215673149428518831237048997865604944110948244509433602537385506616640464500175845009717015495170880109624395554300311270892506142695814694838569, -1721675871351854343451128652352910144490037025118622946896111507342340805427288672947934487232581600784125529888244287409684918807079486233315838804362199288455505004787091729248777728358941730209436946862118156941906336859627208847836437172299959292516730/2541971969072751505856071140988650014521133798955727881776589505057995375638387920504128257341494035439701715736079876129172250864481041861617592684799301706364939710896810664219056265180778785760240497634992032846057171676802264729003)
(-11075950575544307851454446798862191026038143267932569395391423940/13915386198736429269652104583268849980441733321809, 92234245757785275337780676677973797737941499843938183438395648775205576460770931841298359344290/51909027903344538498809167114427100468483556415094026425791603577537884377)
(7501612205052727702200/3286969, 481181295577691400712571864299230/5959274797)
(-19706370502391586232577221444647871986544821890760/24120546710350672082802829608964129, -5627944427593798253125307253104388157267087786776363321230220127118481270/3746111682249571115446035671514914828140431248076367)
(-1348907309573219783845929999561934547241164624786614258441177866755528989367089829/1630420969406675746670906402923592730001782802709147876339266040100, 2105257210730550791191187066456211284686774931722891832226293190641692052726485228284603896487355046249360332421209820711/2081850928262377484771927505797382589849524732025994358498166941999065199414839747390755173980649000)
(-1020486110058820869667605385346303923088522461563202866227519105940/1231549479810945180242465790442089602552976157407769, 39436532235648260375986818990926156966285058942894549911798138181899112078609645823918070143783110/43219306877987062284060221435132245118586093363650979992324190691256410729603)

Commentary

Rank exactly 13, torsion Z/2Z: 13 independent points; 2-isogeny Selmer bound 13, confirmed by a second 2-isogeny descent. Specialisation of Kihara's rank-8 family (S. Kihara, Proc. Japan Acad. 77A (2001) 11-12) with u = (s^2 - l)/(2s), (p,q,s) = (1,9,161/3).

last edited by Alvaro Lozano-Robledo at 2026-09-25 19:16:47 UTC · history