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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 136458j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 136458.bv2 | 136458j1 | \([1, -1, 1, -4400, -142725]\) | \(-7414875/2744\) | \(-3485535231528\) | \([]\) | \(256608\) | \(1.1168\) | \(\Gamma_0(N)\)-optimal |
| 136458.bv3 | 136458j2 | \([1, -1, 1, 33505, 1444231]\) | \(4492125/3584\) | \(-3318798607391232\) | \([]\) | \(769824\) | \(1.6661\) | |
| 136458.bv1 | 136458j3 | \([1, -1, 1, -383450, -91296669]\) | \(-545407363875/14\) | \(-160050087162\) | \([]\) | \(769824\) | \(1.6661\) |
Rank
sage: E.rank()
The elliptic curves in class 136458j have rank \(1\).
Complex multiplication
The elliptic curves in class 136458j do not have complex multiplication.Modular form 136458.2.a.j
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.
\(\left(\begin{array}{rrr} 1 & 3 & 3 \\ 3 & 1 & 9 \\ 3 & 9 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
