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SageMath
E = EllipticCurve("gc1")
E.isogeny_class()
Elliptic curves in class 212160.gc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212160.gc1 | 212160f3 | \([0, 1, 0, -998785, 383865983]\) | \(420339554066191969/244298925\) | \(64041497395200\) | \([2]\) | \(2097152\) | \(1.9739\) | |
| 212160.gc2 | 212160f2 | \([0, 1, 0, -62785, 5909183]\) | \(104413920565969/2472575625\) | \(648170864640000\) | \([2, 2]\) | \(1048576\) | \(1.6273\) | |
| 212160.gc3 | 212160f1 | \([0, 1, 0, -8705, -180225]\) | \(278317173889/109245825\) | \(28638137548800\) | \([2]\) | \(524288\) | \(1.2807\) | \(\Gamma_0(N)\)-optimal |
| 212160.gc4 | 212160f4 | \([0, 1, 0, 7935, 18539775]\) | \(210751100351/566398828125\) | \(-148478054400000000\) | \([2]\) | \(2097152\) | \(1.9739\) |
Rank
sage: E.rank()
The elliptic curves in class 212160.gc have rank \(1\).
Complex multiplication
The elliptic curves in class 212160.gc do not have complex multiplication.Modular form 212160.2.a.gc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
