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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 5712bb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5712.q4 | 5712bb1 | \([0, 1, 0, 16, -6444]\) | \(103823/4386816\) | \(-17968398336\) | \([2]\) | \(4608\) | \(0.64655\) | \(\Gamma_0(N)\)-optimal |
| 5712.q3 | 5712bb2 | \([0, 1, 0, -5104, -139564]\) | \(3590714269297/73410624\) | \(300689915904\) | \([2, 2]\) | \(9216\) | \(0.99312\) | |
| 5712.q1 | 5712bb3 | \([0, 1, 0, -81264, -8943660]\) | \(14489843500598257/6246072\) | \(25583910912\) | \([2]\) | \(18432\) | \(1.3397\) | |
| 5712.q2 | 5712bb4 | \([0, 1, 0, -10864, 226772]\) | \(34623662831857/14438442312\) | \(59139859709952\) | \([4]\) | \(18432\) | \(1.3397\) |
Rank
sage: E.rank()
The elliptic curves in class 5712bb have rank \(1\).
Complex multiplication
The elliptic curves in class 5712bb do not have complex multiplication.Modular form 5712.2.a.bb
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.
