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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 138.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 138.b1 | 138b4 | \([1, 0, 1, -771, 1342]\) | \(50591419971625/28422890688\) | \(28422890688\) | \([2]\) | \(96\) | \(0.69601\) | |
| 138.b2 | 138b2 | \([1, 0, 1, -576, 5266]\) | \(21081759765625/57132\) | \(57132\) | \([6]\) | \(32\) | \(0.14670\) | |
| 138.b3 | 138b1 | \([1, 0, 1, -36, 82]\) | \(-4956477625/268272\) | \(-268272\) | \([6]\) | \(16\) | \(-0.19987\) | \(\Gamma_0(N)\)-optimal |
| 138.b4 | 138b3 | \([1, 0, 1, 189, 190]\) | \(752329532375/448524288\) | \(-448524288\) | \([2]\) | \(48\) | \(0.34944\) |
Rank
sage: E.rank()
The elliptic curves in class 138.b have rank \(0\).
Complex multiplication
The elliptic curves in class 138.b do not have complex multiplication.Modular form 138.2.a.b
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.
