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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 31200.ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31200.ca1 | 31200bu4 | \([0, 1, 0, -17408, -889812]\) | \(72929847752/5265\) | \(42120000000\) | \([2]\) | \(49152\) | \(1.0901\) | |
| 31200.ca2 | 31200bu3 | \([0, 1, 0, -6033, 168063]\) | \(379503424/24375\) | \(1560000000000\) | \([2]\) | \(49152\) | \(1.0901\) | |
| 31200.ca3 | 31200bu1 | \([0, 1, 0, -1158, -12312]\) | \(171879616/38025\) | \(38025000000\) | \([2, 2]\) | \(24576\) | \(0.74348\) | \(\Gamma_0(N)\)-optimal |
| 31200.ca4 | 31200bu2 | \([0, 1, 0, 2592, -72312]\) | \(240641848/428415\) | \(-3427320000000\) | \([2]\) | \(49152\) | \(1.0901\) |
Rank
sage: E.rank()
The elliptic curves in class 31200.ca have rank \(1\).
Complex multiplication
The elliptic curves in class 31200.ca do not have complex multiplication.Modular form 31200.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
