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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 4056.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4056.k1 | 4056j2 | \([0, 1, 0, -5620, 160304]\) | \(34909201168/81\) | \(45556992\) | \([2]\) | \(4608\) | \(0.71250\) | |
| 4056.k2 | 4056j1 | \([0, 1, 0, -355, 2354]\) | \(141150208/6561\) | \(230632272\) | \([2]\) | \(2304\) | \(0.36593\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4056.k have rank \(1\).
Complex multiplication
The elliptic curves in class 4056.k do not have complex multiplication.Modular form 4056.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
