Show commands:
SageMath
E = EllipticCurve("gg1")
E.isogeny_class()
Elliptic curves in class 212160.gg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212160.gg1 | 212160ei4 | \([0, 1, 0, -66945, -6688545]\) | \(126574061279329/16286595\) | \(4269433159680\) | \([2]\) | \(917504\) | \(1.4447\) | |
| 212160.gg2 | 212160ei2 | \([0, 1, 0, -4545, -86625]\) | \(39616946929/10989225\) | \(2880759398400\) | \([2, 2]\) | \(458752\) | \(1.0981\) | |
| 212160.gg3 | 212160ei1 | \([0, 1, 0, -1665, 24543]\) | \(1948441249/89505\) | \(23463198720\) | \([2]\) | \(229376\) | \(0.75153\) | \(\Gamma_0(N)\)-optimal |
| 212160.gg4 | 212160ei3 | \([0, 1, 0, 11775, -553377]\) | \(688699320191/910381875\) | \(-238651146240000\) | \([2]\) | \(917504\) | \(1.4447\) |
Rank
sage: E.rank()
The elliptic curves in class 212160.gg have rank \(1\).
Complex multiplication
The elliptic curves in class 212160.gg do not have complex multiplication.Modular form 212160.2.a.gg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
