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SageMath
sage: E = EllipticCurve("a1")
sage: E.isogeny_class()
Elliptic curves in class 17a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
17.a3 | 17a1 | [1, -1, 1, -1, -14] | [4] | 1 | \(\Gamma_0(N)\)-optimal |
17.a2 | 17a2 | [1, -1, 1, -6, -4] | [2, 2] | 2 | |
17.a1 | 17a3 | [1, -1, 1, -91, -310] | [2] | 4 | |
17.a4 | 17a4 | [1, -1, 1, -1, 0] | [4] | 4 |
Rank
sage: E.rank()
The elliptic curves in class 17a have rank \(0\).
Complex multiplication
The elliptic curves in class 17a do not have complex multiplication.Modular form 17.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.