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SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 12870.v
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12870.v1 | 12870r3 | \([1, -1, 0, -6444, 200070]\) | \(40597630665409/154169730\) | \(112389733170\) | \([2]\) | \(20480\) | \(0.97937\) | |
| 12870.v2 | 12870r2 | \([1, -1, 0, -594, 0]\) | \(31824875809/18404100\) | \(13416588900\) | \([2, 2]\) | \(10240\) | \(0.63280\) | |
| 12870.v3 | 12870r1 | \([1, -1, 0, -414, -3132]\) | \(10779215329/34320\) | \(25019280\) | \([2]\) | \(5120\) | \(0.28622\) | \(\Gamma_0(N)\)-optimal |
| 12870.v4 | 12870r4 | \([1, -1, 0, 2376, -1782]\) | \(2034382787711/1178141250\) | \(-858864971250\) | \([2]\) | \(20480\) | \(0.97937\) |
Rank
sage: E.rank()
The elliptic curves in class 12870.v have rank \(1\).
Complex multiplication
The elliptic curves in class 12870.v do not have complex multiplication.Modular form 12870.2.a.v
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.
