Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 42432k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 42432.r2 | 42432k1 | \([0, -1, 0, -3133, -65555]\) | \(3322336000000/51429573\) | \(52663882752\) | \([2]\) | \(36864\) | \(0.85852\) | \(\Gamma_0(N)\)-optimal |
| 42432.r1 | 42432k2 | \([0, -1, 0, -6193, 86833]\) | \(1603530178000/738501777\) | \(12099613114368\) | \([2]\) | \(73728\) | \(1.2051\) |
Rank
sage: E.rank()
The elliptic curves in class 42432k have rank \(1\).
Complex multiplication
The elliptic curves in class 42432k do not have complex multiplication.Modular form 42432.2.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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
