curve #4237

y² + xy = x³ − 9189067294664874112619890941236188055x + 10727296599988841897085432340806249140932693885977094377

a-invariants: [1, 0, 0, -9189067294664874112619890941236188055, 10727296599988841897085432340806249140932693885977094377]
c-invariants: [441075230143913957405754765179337026641, -9268384262390359399743426387672470193874479665253215081689]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 62659003164479686239328068266541122033367068516956338027806434655324043652257974884230
discriminant (Δ): -53776531496985407486180632129337930670769080170196077225835567106569557337859667173261047054201371172864000000
Faltings height: 19.9971
naive height: 266.9479
primes of bad reduction: 2, 3, 5, 7, 11, 13, 29, 79, 3498445730203303206559439929, 260332218151053659462503699988537659980818327556319
regulator: 1410984072986437583877762.1189554013978907003531185512350672277860087120021144952549299
submitted by: apkorte
submitted at: 2026-10-15 08:08:04 UTC
last updated: 2026-10-15 08:08:04 UTC

Witness: 24 independent points log in to add more points →

(590004994880431594, -2347570066137968070712947047)
(221715824581223254, -2949717622778522433481908587)
(4019127130004479894, -6222360192452655871886312747)
(-165414514065273247514/169, -9521838682300797480254398404359/2197)
(4221539659301752342, -6867970532370704250479515307)
(-935264079507713786, -4301558786706113151917578427)
(-2947621543968607406, 3493259198283447915716669953)
(-107152262846726186, 3422089372621955622976913173)
(2180535375688755094, 1028628759615998185289602453)
(-1437969921314699882, -4579031107805763187416522539)
(1375411824635693014, 830957686596752751086286613)
(1676833536930188464, 183389783049908438952776323)
(14995848968483305651/4, -43057867663770874648012675201/8)
(6771462199853767894, 16093281258301385676838776853)
(-1203332191918229546, -4476869665880211544671289067)
(-3203788854594406706, -2698628632204478746530600347)
(2003085275381266774, -598240466907045954298940267)
(1792345254931721044, 123398865438655023291754903)
(2210148244029665494, -1101875182837009365168954347)
(1759681768750395814, -79257083021118506292684827)
(1910839600114719094, 381479750264758270531510453)
(2646873734002597654, 2224597894463318389922216533)
(9078878610613200599974/7921, -919369750523556388807391608246243/704969)
(1559189441704794454, 436235308750231437199462933)

Commentary

Specialization at T = -284/39 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -9879263860426790634432T^8 + 86542933653687887807616T^7 - 135085452380206943680656T^6 - 670355493830703722869488T^5 - 1051946747940271972123080T^4 - 3373640011473668508665088T^3 - 849884467368517694646336T^2 + 3646401529809912988876056T - 1342215706985392840165467, A6 = 376157002259901169222608370611456T^12 - 5012568105195271642968657466363392T^11 + 20939308987938975132973897671040896T^10 + 13327362718284003523861442478906240T^9 - 278883358041208378819482351125270640T^8 + 367979789534049131266298576914822080T^7 + 2889329182749276846125866347500564400T^6 + 1554211227618089538131391252658550640T^5 - 3013942383637349393404424222146644840T^4 + 631952356094086111413853506833246400T^3 + 2160468957381686900407596405190211280T^2 - 2226612923028643159215506958258590280T + 628996598302785742517854074617969910. It is a 2-neighbour of another fibration built here (itself a 2-neighbour of the fibration 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 08:08:04 UTC · history