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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 10780f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10780.k4 | 10780f1 | \([0, -1, 0, -2221, -38730]\) | \(643956736/15125\) | \(28471058000\) | \([2]\) | \(10368\) | \(0.79153\) | \(\Gamma_0(N)\)-optimal |
| 10780.k3 | 10780f2 | \([0, -1, 0, -4916, 76616]\) | \(436334416/171875\) | \(5176556000000\) | \([2]\) | \(20736\) | \(1.1381\) | |
| 10780.k2 | 10780f3 | \([0, -1, 0, -21821, 1232330]\) | \(610462990336/8857805\) | \(16673790407120\) | \([2]\) | \(31104\) | \(1.3408\) | |
| 10780.k1 | 10780f4 | \([0, -1, 0, -347916, 79103816]\) | \(154639330142416/33275\) | \(1002181241600\) | \([2]\) | \(62208\) | \(1.6874\) |
Rank
sage: E.rank()
The elliptic curves in class 10780f have rank \(1\).
Complex multiplication
The elliptic curves in class 10780f do not have complex multiplication.Modular form 10780.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
