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SageMath
E = EllipticCurve("gb1")
E.isogeny_class()
Elliptic curves in class 227136.gb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 227136.gb1 | 227136j3 | \([0, 1, 0, -865089, -309876033]\) | \(124318741396429/51631104\) | \(29735920726966272\) | \([2]\) | \(2304000\) | \(2.1228\) | |
| 227136.gb2 | 227136j4 | \([0, 1, 0, -731969, -408358209]\) | \(-75306487574989/81352871712\) | \(-46853589342948950016\) | \([2]\) | \(4608000\) | \(2.4693\) | |
| 227136.gb3 | 227136j1 | \([0, 1, 0, -28929, 1881855]\) | \(4649101309/6804\) | \(3918630223872\) | \([2]\) | \(460800\) | \(1.3180\) | \(\Gamma_0(N)\)-optimal |
| 227136.gb4 | 227136j2 | \([0, 1, 0, -20609, 2995071]\) | \(-1680914269/5786802\) | \(-3332795005403136\) | \([2]\) | \(921600\) | \(1.6646\) |
Rank
sage: E.rank()
The elliptic curves in class 227136.gb have rank \(1\).
Complex multiplication
The elliptic curves in class 227136.gb do not have complex multiplication.Modular form 227136.2.a.gb
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
