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SageMath
sage: E = EllipticCurve("105.a1")
sage: E.isogeny_class()
Elliptic curves in class 105a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
105.a3 | 105a1 | [1, 0, 1, -3, 1] | [2] | 4 | \(\Gamma_0(N)\)-optimal |
105.a2 | 105a2 | [1, 0, 1, -8, -7] | [2, 2] | 8 | |
105.a1 | 105a3 | [1, 0, 1, -113, -469] | [2] | 16 | |
105.a4 | 105a4 | [1, 0, 1, 17, -37] | [4] | 16 |
Rank
sage: E.rank()
The elliptic curves in class 105a have rank \(0\).
Modular form 105.2.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.