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SageMath
E = EllipticCurve("dg1")
E.isogeny_class()
Elliptic curves in class 12480.dg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12480.dg1 | 12480cz3 | \([0, 1, 0, -6625, 204575]\) | \(490757540836/2142075\) | \(140383027200\) | \([2]\) | \(24576\) | \(0.99189\) | |
| 12480.dg2 | 12480cz2 | \([0, 1, 0, -625, -625]\) | \(1650587344/950625\) | \(15575040000\) | \([2, 2]\) | \(12288\) | \(0.64532\) | |
| 12480.dg3 | 12480cz1 | \([0, 1, 0, -445, -3757]\) | \(9538484224/26325\) | \(26956800\) | \([2]\) | \(6144\) | \(0.29875\) | \(\Gamma_0(N)\)-optimal |
| 12480.dg4 | 12480cz4 | \([0, 1, 0, 2495, -2497]\) | \(26198797244/15234375\) | \(-998400000000\) | \([2]\) | \(24576\) | \(0.99189\) |
Rank
sage: E.rank()
The elliptic curves in class 12480.dg have rank \(0\).
Complex multiplication
The elliptic curves in class 12480.dg do not have complex multiplication.Modular form 12480.2.a.dg
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.
