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SageMath
E = EllipticCurve("fa1")
E.isogeny_class()
Elliptic curves in class 155610.fa
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 155610.fa1 | 155610f7 | \([1, -1, 1, -119815963367, 15963236505248159]\) | \(260939746299651996897062684320310569/4502310731746516416000000\) | \(3282184523443210467264000000\) | \([12]\) | \(509607936\) | \(4.8211\) | |
| 155610.fa2 | 155610f6 | \([1, -1, 1, -7495963367, 248904729248159]\) | \(63896717795469435410864800310569/264596810010624000000000000\) | \(192891074497744896000000000000\) | \([2, 6]\) | \(254803968\) | \(4.4745\) | |
| 155610.fa3 | 155610f8 | \([1, -1, 1, -3892323047, 488660685562271]\) | \(-8945874824846901999181300606249/136327175537109375000000000000\) | \(-99382510966552734375000000000000\) | \([6]\) | \(509607936\) | \(4.8211\) | |
| 155610.fa4 | 155610f4 | \([1, -1, 1, -1571158832, 19021614439331]\) | \(588378042102789360957899335609/126092479227795814426383600\) | \(91921417357063148716833644400\) | \([4]\) | \(169869312\) | \(4.2718\) | |
| 155610.fa5 | 155610f3 | \([1, -1, 1, -701190887, -379165589089]\) | \(52300395461270777993673352489/30181158775846600704000000\) | \(22002064747592171913216000000\) | \([6]\) | \(127401984\) | \(4.1279\) | |
| 155610.fa6 | 155610f2 | \([1, -1, 1, -503416832, -4087739215069]\) | \(19354385182631020519933543609/1297852417956067777440000\) | \(946134412689973409753760000\) | \([2, 2]\) | \(84934656\) | \(3.9252\) | |
| 155610.fa7 | 155610f1 | \([1, -1, 1, -495018752, -4239025587421]\) | \(18401835147394456911544300729/91846771642205798400\) | \(66956296527168027033600\) | \([2]\) | \(42467328\) | \(3.5786\) | \(\Gamma_0(N)\)-optimal |
| 155610.fa8 | 155610f5 | \([1, -1, 1, 429955888, -17514865815901]\) | \(12057794750690459080173411911/188765599666333542693750000\) | \(-137610122156757152623743750000\) | \([2]\) | \(169869312\) | \(4.2718\) |
Rank
sage: E.rank()
The elliptic curves in class 155610.fa have rank \(1\).
Complex multiplication
The elliptic curves in class 155610.fa do not have complex multiplication.Modular form 155610.2.a.fa
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
