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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 32856.g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 32856.g1 | 32856c6 | \([0, -1, 0, -526152, -146722068]\) | \(3065617154/9\) | \(47291469170688\) | \([2]\) | \(202752\) | \(1.8533\) | |
| 32856.g2 | 32856c4 | \([0, -1, 0, -88072, 10088668]\) | \(28756228/3\) | \(7881911528448\) | \([2]\) | \(101376\) | \(1.5067\) | |
| 32856.g3 | 32856c3 | \([0, -1, 0, -33312, -2221380]\) | \(1556068/81\) | \(212811611268096\) | \([2, 2]\) | \(101376\) | \(1.5067\) | |
| 32856.g4 | 32856c2 | \([0, -1, 0, -5932, 133300]\) | \(35152/9\) | \(5911433646336\) | \([2, 2]\) | \(50688\) | \(1.1601\) | |
| 32856.g5 | 32856c1 | \([0, -1, 0, 913, 12828]\) | \(2048/3\) | \(-123154867632\) | \([2]\) | \(25344\) | \(0.81353\) | \(\Gamma_0(N)\)-optimal |
| 32856.g6 | 32856c5 | \([0, -1, 0, 21448, -8858292]\) | \(207646/6561\) | \(-34475481025431552\) | \([2]\) | \(202752\) | \(1.8533\) |
Rank
sage: E.rank()
The elliptic curves in class 32856.g have rank \(1\).
Complex multiplication
The elliptic curves in class 32856.g do not have complex multiplication.Modular form 32856.2.a.g
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
