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SageMath
sage: E = EllipticCurve("a1")
sage: E.isogeny_class()
Elliptic curves in class 49.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality | CM discriminant |
---|---|---|---|---|---|---|
49.a1 | 49a4 | [1, -1, 0, -1822, 30393] | [2] | 14 | -28 | |
49.a2 | 49a3 | [1, -1, 0, -107, 552] | [2] | 7 | -7 | |
49.a3 | 49a2 | [1, -1, 0, -37, -78] | [2] | 2 | -28 | |
49.a4 | 49a1 | [1, -1, 0, -2, -1] | [2] | 1 | \(\Gamma_0(N)\)-optimal | -7 |
Rank
sage: E.rank()
The elliptic curves in class 49.a have rank \(0\).
Complex multiplication
Each elliptic curve in class 49.a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-7}) \).Modular form 49.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.