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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 136710.ba
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 136710.ba1 | 136710fa4 | \([1, -1, 0, -902295, 144201721]\) | \(947226559343329/443751840500\) | \(38058874046295700500\) | \([2]\) | \(3981312\) | \(2.4515\) | |
| 136710.ba2 | 136710fa2 | \([1, -1, 0, -752355, 251366485]\) | \(549131937598369/307520\) | \(26374797529920\) | \([2]\) | \(1327104\) | \(1.9022\) | |
| 136710.ba3 | 136710fa1 | \([1, -1, 0, -46755, 3983125]\) | \(-131794519969/3174400\) | \(-272255974502400\) | \([2]\) | \(663552\) | \(1.5556\) | \(\Gamma_0(N)\)-optimal |
| 136710.ba4 | 136710fa3 | \([1, -1, 0, 200205, 16973221]\) | \(10347405816671/7447750000\) | \(-638764627677750000\) | \([2]\) | \(1990656\) | \(2.1049\) |
Rank
sage: E.rank()
The elliptic curves in class 136710.ba have rank \(1\).
Complex multiplication
The elliptic curves in class 136710.ba do not have complex multiplication.Modular form 136710.2.a.ba
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
