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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 91728z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 91728.bo3 | 91728z1 | \([0, 0, 0, -17346, -879109]\) | \(420616192/117\) | \(160554178512\) | \([2]\) | \(147456\) | \(1.1319\) | \(\Gamma_0(N)\)-optimal |
| 91728.bo2 | 91728z2 | \([0, 0, 0, -19551, -641410]\) | \(37642192/13689\) | \(300557422174464\) | \([2, 2]\) | \(294912\) | \(1.4785\) | |
| 91728.bo4 | 91728z3 | \([0, 0, 0, 59829, -4531030]\) | \(269676572/257049\) | \(-22575201932215296\) | \([2]\) | \(589824\) | \(1.8251\) | |
| 91728.bo1 | 91728z4 | \([0, 0, 0, -134211, 18460946]\) | \(3044193988/85293\) | \(7490815752655872\) | \([2]\) | \(589824\) | \(1.8251\) |
Rank
sage: E.rank()
The elliptic curves in class 91728z have rank \(1\).
Complex multiplication
The elliptic curves in class 91728z do not have complex multiplication.Modular form 91728.2.a.z
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.
