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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 221778.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221778.q1 | 221778c3 | \([1, -1, 1, -1474670, 689640445]\) | \(-189613868625/128\) | \(-239413062676608\) | \([]\) | \(2159136\) | \(2.0748\) | |
| 221778.q2 | 221778c4 | \([1, -1, 1, -1166645, 985541581]\) | \(-1159088625/2097152\) | \(-317726033130377183232\) | \([]\) | \(6477408\) | \(2.6241\) | |
| 221778.q3 | 221778c2 | \([1, -1, 1, -57755, -5584301]\) | \(-140625/8\) | \(-1212028629800328\) | \([]\) | \(925344\) | \(1.6512\) | |
| 221778.q4 | 221778c1 | \([1, -1, 1, 3850, -15209]\) | \(3375/2\) | \(-3740829104322\) | \([]\) | \(308448\) | \(1.1019\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 221778.q have rank \(1\).
Complex multiplication
The elliptic curves in class 221778.q do not have complex multiplication.Modular form 221778.2.a.q
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 21 & 7 \\ 3 & 1 & 7 & 21 \\ 21 & 7 & 1 & 3 \\ 7 & 21 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
