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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 5850.ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5850.ca1 | 5850bd4 | \([1, -1, 1, -113105, 8797897]\) | \(520300455507/193072360\) | \(59378800966875000\) | \([2]\) | \(82944\) | \(1.9185\) | |
| 5850.ca2 | 5850bd2 | \([1, -1, 1, -99980, 12192897]\) | \(261984288445803/42250\) | \(17824218750\) | \([2]\) | \(27648\) | \(1.3692\) | |
| 5850.ca3 | 5850bd1 | \([1, -1, 1, -6230, 192897]\) | \(-63378025803/812500\) | \(-342773437500\) | \([2]\) | \(13824\) | \(1.0226\) | \(\Gamma_0(N)\)-optimal |
| 5850.ca4 | 5850bd3 | \([1, -1, 1, 21895, 967897]\) | \(3774555693/3515200\) | \(-1081088775000000\) | \([2]\) | \(41472\) | \(1.5720\) |
Rank
sage: E.rank()
The elliptic curves in class 5850.ca have rank \(0\).
Complex multiplication
The elliptic curves in class 5850.ca do not have complex multiplication.Modular form 5850.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
