Analysis of Elliptic Curve #273: A Rank $\ge 30$ Milestone

This document details the properties of Curve #273, a significant entry in the Elliptic Curve Rank Leaderboard. This specific curve is notable for its high rank and its status as a record-holder across several mathematical metrics.

📐 Mathematical Definition

The curve is defined by the following Weierstrass equation:

$y^2 + xy = x^3 + Ax + B$

Where the coefficients are:

• A = 201769035260418549083594900060734240952308696994802735114305555
• B = 1151107939141058565733479426024323225135665982951300586808823640527729578307228357301072889377

a-invariants

The curve's properties are summarized by its a-invariants $[a_1, a_2, a_3, a_4, a_6]$:

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📊 Key Technical Metrics

The curve exhibits extreme values across several fundamental elliptic curve parameters:

• Rank (Lower Bound): $30$
• Conductor ($N$): 2381958488309327728488641214562148525681586925734398576894288277390640894675828305511266759053188773997283310012808983216600083938367948090232306090
• Discriminant ($\Delta$): -467146612553087673145676887338415319189833560021597726132568408428516502540365187011005783426015535135792222727102204968876160345269834928439540905541970336381372450377910440530176000000000
• Regulator: 10720686604115654188358042985326663938.588084900092986474567459949780498792191278825612795832355129

🏆 Record-Breaking Status

Curve #273 currently holds the record for rank 30 in the following categories:

• Naive Height: $442.0854$
• Faltings Height: $34.7705$
• Discriminant ($\Delta$)

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🛠️ Reduction and Submission Data

Primes of Bad Reduction

The curve has bad reduction at the following primes: 2, 3, 5, 7, 13, 31, 41, 47, 53, 67, 379, 4349, 25721454817, 97018222656318846556561979214040553412450110580812087282349817173780902099339117104673990259247421230916714670243202937

Submission Timeline

• Submitted by: ranksunbounded
• Submission Date: 2026-08-20 08:10:53 UTC
• Last Update: 2026-08-20 11:24:48 UTC

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📍 Witness Points

To prove the rank is at least 30, the following 30 independent points were provided:

# Independent points (x, y)
points = [
    (-4761204159891138283979053265906, 44764265461782973805868732003346421827415264953),
    (-14158422539541566469588779426546, 34199834254251713784176619895082644508395077433),
    (-11522667358396562420423130332066, -44115070023357103726405378140637465204943359607),
    (-204839531927226269712122049566, -34531574232452693997231136031282772551453427107),
    (3899324051227528532535432912094, 20582352852872417675268569815574934013539218953),
    (149851368287976334870008075289384, -1826442728148288630645637436047625928557963231657),
    (240440240734591134232325971191694, 3721941824016160691689265341458606456425791434553),
    (58446054919170749975942104376446/9, -289145377197241504032247540119122580900747897469/27),
    (25642661602146479458845459929344, -113306861325798987289137854129016658652160209297),
    (25720885078613923889202869994094, -113918565504468051036791617945007239588074855047),
    (4956414590296956229584100339596814, 348939117745197814060339374812186839746231405619513),
    (725964821994104294477684670330094, -19556488133953913131900670560396205869420775943047),
    (20802191136944676997135829374, 33866070189878993817062821320678356972094522793),
    (79052318332408565020526148386446/9, -202982221452031541387733280916787231176177841869/27),
    (-11232245340662775388535509780886, -44725045659073489550941507272219743508825024527),
    (8362456338772815315335239525614, -6972475614865802969141741730862843401376795527),
    (2011658715643038193607509024534, 27447371869432010931671648582500375378543228993),
    (5027695440284894460797358334207726/529, -116678851641395817353208818767411586490893208148849/12167),
    (24649144267565165528439068441554, 105612540318783792474731275264940719325335867213),
    (-12211389420609043025008816968566, 42356225616159991618318584560811156010370207173),
    (7798692390172953821075781106768126/1369, -691870045568822811690292896396241871567834072004011/50653),
    (871579928157