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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 234135bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 234135.bg4 | 234135bg1 | \([1, -1, 0, 1611, 226408]\) | \(357911/17415\) | \(-22490914680135\) | \([2]\) | \(450560\) | \(1.2418\) | \(\Gamma_0(N)\)-optimal |
| 234135.bg3 | 234135bg2 | \([1, -1, 0, -47394, 3823375]\) | \(9116230969/416025\) | \(537282961803225\) | \([2, 2]\) | \(901120\) | \(1.5883\) | |
| 234135.bg1 | 234135bg3 | \([1, -1, 0, -749799, 250086568]\) | \(36097320816649/80625\) | \(104124605000625\) | \([2]\) | \(1802240\) | \(1.9349\) | |
| 234135.bg2 | 234135bg4 | \([1, -1, 0, -129069, -12821990]\) | \(184122897769/51282015\) | \(66229079758277535\) | \([2]\) | \(1802240\) | \(1.9349\) |
Rank
sage: E.rank()
The elliptic curves in class 234135bg have rank \(0\).
Complex multiplication
The elliptic curves in class 234135bg do not have complex multiplication.Modular form 234135.2.a.bg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
