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SageMath
sage: E = EllipticCurve("a1")
sage: E.isogeny_class()
Elliptic curves in class 15.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
15.a1 | 15a5 | [1, 1, 1, -2160, -39540] | [2] | 4 | |
15.a2 | 15a2 | [1, 1, 1, -135, -660] | [2, 2] | 2 | |
15.a3 | 15a6 | [1, 1, 1, -110, -880] | [2] | 4 | |
15.a4 | 15a7 | [1, 1, 1, -80, 242] | [4] | 4 | |
15.a5 | 15a1 | [1, 1, 1, -10, -10] | [2, 4] | 1 | \(\Gamma_0(N)\)-optimal |
15.a6 | 15a3 | [1, 1, 1, -5, 2] | [2, 4] | 2 | |
15.a7 | 15a8 | [1, 1, 1, 0, 0] | [4] | 4 | |
15.a8 | 15a4 | [1, 1, 1, 35, -28] | [8] | 2 |
Rank
sage: E.rank()
The elliptic curves in class 15.a have rank \(0\).
Complex multiplication
The elliptic curves in class 15.a do not have complex multiplication.Modular form 15.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.