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SageMath
E = EllipticCurve("gk1")
E.isogeny_class()
Elliptic curves in class 262080gk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 262080.gk4 | 262080gk1 | \([0, 0, 0, 350232, -86710808]\) | \(6364491337435136/8034291412875\) | \(-5997566402545536000\) | \([2]\) | \(5898240\) | \(2.2892\) | \(\Gamma_0(N)\)-optimal |
| 262080.gk3 | 262080gk2 | \([0, 0, 0, -2113788, -838729712]\) | \(87450143958975184/25164018140625\) | \(300557422174464000000\) | \([2, 2]\) | \(11796480\) | \(2.6357\) | |
| 262080.gk2 | 262080gk3 | \([0, 0, 0, -12643788, 16645282288]\) | \(4678944235881273796/202428825314625\) | \(9671187736452243456000\) | \([2]\) | \(23592960\) | \(2.9823\) | |
| 262080.gk1 | 262080gk4 | \([0, 0, 0, -31008108, -66451951568]\) | \(69014771940559650916/9797607421875\) | \(468087984000000000000\) | \([2]\) | \(23592960\) | \(2.9823\) |
Rank
sage: E.rank()
The elliptic curves in class 262080gk have rank \(2\).
Complex multiplication
The elliptic curves in class 262080gk do not have complex multiplication.Modular form 262080.2.a.gk
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.
