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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 24640z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 24640.q4 | 24640z1 | \([0, 0, 0, -188, -2448]\) | \(-44851536/132055\) | \(-2163589120\) | \([2]\) | \(8192\) | \(0.47816\) | \(\Gamma_0(N)\)-optimal |
| 24640.q3 | 24640z2 | \([0, 0, 0, -4108, -101232]\) | \(116986321764/148225\) | \(9714073600\) | \([2, 2]\) | \(16384\) | \(0.82474\) | |
| 24640.q2 | 24640z3 | \([0, 0, 0, -5228, -41648]\) | \(120564797922/64054375\) | \(8395735040000\) | \([2]\) | \(32768\) | \(1.1713\) | |
| 24640.q1 | 24640z4 | \([0, 0, 0, -65708, -6482992]\) | \(239369344910082/385\) | \(50462720\) | \([2]\) | \(32768\) | \(1.1713\) |
Rank
sage: E.rank()
The elliptic curves in class 24640z have rank \(0\).
Complex multiplication
The elliptic curves in class 24640z do not have complex multiplication.Modular form 24640.2.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.
