Show commands:
SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 28322.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 28322.r1 | 28322ba4 | \([1, 0, 0, -1600488, 538413406]\) | \(159661140625/48275138\) | \(137089847175688303778\) | \([2]\) | \(995328\) | \(2.5690\) | |
| 28322.r2 | 28322ba3 | \([1, 0, 0, -1458878, 678012544]\) | \(120920208625/19652\) | \(55806980327982212\) | \([2]\) | \(497664\) | \(2.2225\) | |
| 28322.r3 | 28322ba2 | \([1, 0, 0, -609218, -183032900]\) | \(8805624625/2312\) | \(6565527097409672\) | \([2]\) | \(331776\) | \(2.0197\) | |
| 28322.r4 | 28322ba1 | \([1, 0, 0, -42778, -2111964]\) | \(3048625/1088\) | \(3089659810545728\) | \([2]\) | \(165888\) | \(1.6732\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 28322.r have rank \(1\).
Complex multiplication
The elliptic curves in class 28322.r do not have complex multiplication.Modular form 28322.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
