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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 63504bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 63504.bo3 | 63504bg1 | \([0, 0, 0, -3675, 89866]\) | \(-140625/8\) | \(-312264916992\) | \([]\) | \(51840\) | \(0.96251\) | \(\Gamma_0(N)\)-optimal |
| 63504.bo4 | 63504bg2 | \([0, 0, 0, 19845, 166698]\) | \(3375/2\) | \(-512192530096128\) | \([]\) | \(155520\) | \(1.5118\) | |
| 63504.bo2 | 63504bg3 | \([0, 0, 0, -74235, -15814358]\) | \(-1159088625/2097152\) | \(-81858374399950848\) | \([]\) | \(362880\) | \(1.9355\) | |
| 63504.bo1 | 63504bg4 | \([0, 0, 0, -7600635, -8065349334]\) | \(-189613868625/128\) | \(-32780321926152192\) | \([]\) | \(1088640\) | \(2.4848\) |
Rank
sage: E.rank()
The elliptic curves in class 63504bg have rank \(1\).
Complex multiplication
The elliptic curves in class 63504bg do not have complex multiplication.Modular form 63504.2.a.bg
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 7 & 21 \\ 3 & 1 & 21 & 7 \\ 7 & 21 & 1 & 3 \\ 21 & 7 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
