Show commands:
SageMath
sage: E = EllipticCurve("dk1")
sage: E.isogeny_class()
Elliptic curves in class 123840.dk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 123840.dk1 | 123840cd4 | \([0, 0, 0, -482988, -128316112]\) | \(65202655558249/512820150\) | \(98001456817766400\) | \([2]\) | \(1179648\) | \(2.0885\) | |
| 123840.dk2 | 123840cd2 | \([0, 0, 0, -50988, 1111088]\) | \(76711450249/41602500\) | \(7950361559040000\) | \([2, 2]\) | \(589824\) | \(1.7419\) | |
| 123840.dk3 | 123840cd1 | \([0, 0, 0, -39468, 3014192]\) | \(35578826569/51600\) | \(9860913561600\) | \([2]\) | \(294912\) | \(1.3954\) | \(\Gamma_0(N)\)-optimal |
| 123840.dk4 | 123840cd3 | \([0, 0, 0, 196692, 8739632]\) | \(4403686064471/2721093750\) | \(-520009113600000000\) | \([2]\) | \(1179648\) | \(2.0885\) |
Rank
sage: E.rank()
The elliptic curves in class 123840.dk have rank \(1\).
Complex multiplication
The elliptic curves in class 123840.dk do not have complex multiplication.Modular form 123840.2.a.dk
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.
