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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 414960x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414960.x7 | 414960x1 | \([0, -1, 0, -880033336, -10048060651664]\) | \(18401835147394456911544300729/91846771642205798400\) | \(376204376646474950246400\) | \([2]\) | \(127401984\) | \(3.7225\) | \(\Gamma_0(N)\)-optimal* |
414960.x6 | 414960x2 | \([0, -1, 0, -894963256, -9689455917200]\) | \(19354385182631020519933543609/1297852417956067777440000\) | \(5316003503948053616394240000\) | \([2, 2]\) | \(254803968\) | \(4.0691\) | \(\Gamma_0(N)\)-optimal* |
414960.x5 | 414960x3 | \([0, -1, 0, -1246561576, -898762877840]\) | \(52300395461270777993673352489/30181158775846600704000000\) | \(123622026345867676483584000000\) | \([2]\) | \(382205952\) | \(4.2718\) | \(\Gamma_0(N)\)-optimal* |
414960.x4 | 414960x4 | \([0, -1, 0, -2793171256, 45088271263600]\) | \(588378042102789360957899335609/126092479227795814426383600\) | \(516474794917051655890467225600\) | \([2]\) | \(509607936\) | \(4.4156\) | \(\Gamma_0(N)\)-optimal* |
414960.x8 | 414960x5 | \([0, -1, 0, 764366024, -41516718971024]\) | \(12057794750690459080173411911/188765599666333542693750000\) | \(-773183896233302190873600000000\) | \([2]\) | \(509607936\) | \(4.4156\) | |
414960.x2 | 414960x6 | \([0, -1, 0, -13326157096, 589996395254896]\) | \(63896717795469435410864800310569/264596810010624000000000000\) | \(1083788533803515904000000000000\) | \([2, 2]\) | \(764411904\) | \(4.6184\) | \(\Gamma_0(N)\)-optimal* |
414960.x1 | 414960x7 | \([0, -1, 0, -213006157096, 37838782827254896]\) | \(260939746299651996897062684320310569/4502310731746516416000000\) | \(18441464757233731239936000000\) | \([2]\) | \(1528823808\) | \(4.9649\) | \(\Gamma_0(N)\)-optimal* |
414960.x3 | 414960x8 | \([0, -1, 0, -6919685416, 1158306810221680]\) | \(-8945874824846901999181300606249/136327175537109375000000000000\) | \(-558396111000000000000000000000000\) | \([2]\) | \(1528823808\) | \(4.9649\) |
Rank
sage: E.rank()
The elliptic curves in class 414960x have rank \(1\).
Complex multiplication
The elliptic curves in class 414960x do not have complex multiplication.Modular form 414960.2.a.x
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.