Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 80688.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 80688.k1 | 80688q2 | \([0, -1, 0, -8385388, 9348947308]\) | \(31899394000/3\) | \(6132422575964928\) | \([]\) | \(1983744\) | \(2.4647\) | |
| 80688.k2 | 80688q1 | \([0, -1, 0, -114868, 9876124]\) | \(82000/27\) | \(55191803183684352\) | \([]\) | \(661248\) | \(1.9154\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 80688.k have rank \(1\).
Complex multiplication
The elliptic curves in class 80688.k do not have complex multiplication.Modular form 80688.2.a.k
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
