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SageMath
E = EllipticCurve("er1")
E.isogeny_class()
Elliptic curves in class 215600.er
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 215600.er1 | 215600gz3 | \([0, 0, 0, -307475, 64741250]\) | \(1707831108/26411\) | \(49715643824000000\) | \([2]\) | \(1572864\) | \(2.0051\) | |
| 215600.er2 | 215600gz2 | \([0, 0, 0, -37975, -1286250]\) | \(12869712/5929\) | \(2790163684000000\) | \([2, 2]\) | \(786432\) | \(1.6585\) | |
| 215600.er3 | 215600gz1 | \([0, 0, 0, -31850, -2186625]\) | \(121485312/77\) | \(2264743250000\) | \([2]\) | \(393216\) | \(1.3119\) | \(\Gamma_0(N)\)-optimal |
| 215600.er4 | 215600gz4 | \([0, 0, 0, 133525, -9689750]\) | \(139863132/102487\) | \(-192919889008000000\) | \([2]\) | \(1572864\) | \(2.0051\) |
Rank
sage: E.rank()
The elliptic curves in class 215600.er have rank \(1\).
Complex multiplication
The elliptic curves in class 215600.er do not have complex multiplication.Modular form 215600.2.a.er
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.
