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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 63504.bi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 63504.bi1 | 63504co3 | \([0, 0, 0, -844515, 298716642]\) | \(-189613868625/128\) | \(-44966148046848\) | \([]\) | \(362880\) | \(1.9355\) | |
| 63504.bi2 | 63504co4 | \([0, 0, 0, -668115, 426987666]\) | \(-1159088625/2097152\) | \(-59674754937564168192\) | \([]\) | \(1088640\) | \(2.4848\) | |
| 63504.bi3 | 63504co2 | \([0, 0, 0, -33075, -2426382]\) | \(-140625/8\) | \(-227641124487168\) | \([]\) | \(155520\) | \(1.5118\) | |
| 63504.bi4 | 63504co1 | \([0, 0, 0, 2205, -6174]\) | \(3375/2\) | \(-702596063232\) | \([]\) | \(51840\) | \(0.96251\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 63504.bi have rank \(2\).
Complex multiplication
The elliptic curves in class 63504.bi do not have complex multiplication.Modular form 63504.2.a.bi
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.
