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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 54080.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 54080.k1 | 54080cj1 | \([0, 1, 0, -641, -6785]\) | \(-658489/40\) | \(-1772093440\) | \([]\) | \(27648\) | \(0.52951\) | \(\Gamma_0(N)\)-optimal |
| 54080.k2 | 54080cj2 | \([0, 1, 0, 3519, -9281]\) | \(108750551/64000\) | \(-2835349504000\) | \([]\) | \(82944\) | \(1.0788\) |
Rank
sage: E.rank()
The elliptic curves in class 54080.k have rank \(2\).
Complex multiplication
The elliptic curves in class 54080.k do not have complex multiplication.Modular form 54080.2.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.
