Show commands:
SageMath
E = EllipticCurve("ed1")
E.isogeny_class()
Elliptic curves in class 64350ed
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 64350.dz4 | 64350ed1 | \([1, -1, 1, 246145, 101428647]\) | \(144794100308831/474439680000\) | \(-5404164480000000000\) | \([2]\) | \(1179648\) | \(2.2776\) | \(\Gamma_0(N)\)-optimal |
| 64350.dz3 | 64350ed2 | \([1, -1, 1, -2345855, 1195252647]\) | \(125337052492018849/18404100000000\) | \(209634201562500000000\) | \([2, 2]\) | \(2359296\) | \(2.6242\) | |
| 64350.dz2 | 64350ed3 | \([1, -1, 1, -10067855, -11113615353]\) | \(9908022260084596129/1047363281250000\) | \(11930122375488281250000\) | \([2]\) | \(4718592\) | \(2.9708\) | |
| 64350.dz1 | 64350ed4 | \([1, -1, 1, -36095855, 83477752647]\) | \(456612868287073618849/12544848030000\) | \(142893659591718750000\) | \([2]\) | \(4718592\) | \(2.9708\) |
Rank
sage: E.rank()
The elliptic curves in class 64350ed have rank \(0\).
Complex multiplication
The elliptic curves in class 64350ed do not have complex multiplication.Modular form 64350.2.a.ed
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.
