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SageMath
E = EllipticCurve("cg1")
E.isogeny_class()
Elliptic curves in class 127296cg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 127296.u4 | 127296cg1 | \([0, 0, 0, -310476, 66587024]\) | \(17319700013617/25857\) | \(4941349650432\) | \([2]\) | \(524288\) | \(1.7045\) | \(\Gamma_0(N)\)-optimal |
| 127296.u3 | 127296cg2 | \([0, 0, 0, -313356, 65288720]\) | \(17806161424897/668584449\) | \(127768477911220224\) | \([2, 2]\) | \(1048576\) | \(2.0510\) | |
| 127296.u5 | 127296cg3 | \([0, 0, 0, 127284, 234318224]\) | \(1193377118543/124806800313\) | \(-23850950964852031488\) | \([2]\) | \(2097152\) | \(2.3976\) | |
| 127296.u2 | 127296cg4 | \([0, 0, 0, -800076, -186832240]\) | \(296380748763217/92608836489\) | \(17697824256945291264\) | \([2, 2]\) | \(2097152\) | \(2.3976\) | |
| 127296.u6 | 127296cg5 | \([0, 0, 0, 2232564, -1265239024]\) | \(6439735268725823/7345472585373\) | \(-1403741671191194370048\) | \([2]\) | \(4194304\) | \(2.7442\) | |
| 127296.u1 | 127296cg6 | \([0, 0, 0, -11620236, -15244166896]\) | \(908031902324522977/161726530797\) | \(30906421333462351872\) | \([2]\) | \(4194304\) | \(2.7442\) |
Rank
sage: E.rank()
The elliptic curves in class 127296cg have rank \(1\).
Complex multiplication
The elliptic curves in class 127296cg do not have complex multiplication.Modular form 127296.2.a.cg
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
