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SageMath
sage: E = EllipticCurve("k1")
sage: E.isogeny_class()
Elliptic curves in class 53312.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
53312.k1 | 53312ba4 | [0, 1, 0, -354433, 56005599] | [2] | 663552 | |
53312.k2 | 53312ba3 | [0, 1, 0, -323073, 70562911] | [2] | 331776 | |
53312.k3 | 53312ba2 | [0, 1, 0, -134913, -19114145] | [2] | 221184 | |
53312.k4 | 53312ba1 | [0, 1, 0, -9473, -222881] | [2] | 110592 | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 53312.k have rank \(1\).
Complex multiplication
The elliptic curves in class 53312.k do not have complex multiplication.Modular form 53312.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.