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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 134640ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 134640.bm3 | 134640ca1 | \([0, 0, 0, -7542363, -7971324662]\) | \(15891267085572193561/3334993530000\) | \(9958237320683520000\) | \([2]\) | \(3538944\) | \(2.6414\) | \(\Gamma_0(N)\)-optimal |
| 134640.bm2 | 134640ca2 | \([0, 0, 0, -8374683, -6103432118]\) | \(21754112339458491481/7199734626562500\) | \(21498292399161600000000\) | \([2, 2]\) | \(7077888\) | \(2.9880\) | |
| 134640.bm1 | 134640ca3 | \([0, 0, 0, -54274683, 149341507882]\) | \(5921450764096952391481/200074809015963750\) | \(597420178524723502080000\) | \([4]\) | \(14155776\) | \(3.3346\) | |
| 134640.bm4 | 134640ca4 | \([0, 0, 0, 24208197, -42003249302]\) | \(525440531549759128199/559322204589843750\) | \(-1670127153750000000000000\) | \([2]\) | \(14155776\) | \(3.3346\) |
Rank
sage: E.rank()
The elliptic curves in class 134640ca have rank \(1\).
Complex multiplication
The elliptic curves in class 134640ca do not have complex multiplication.Modular form 134640.2.a.ca
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
