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SageMath
sage: E = EllipticCurve("a1")
sage: E.isogeny_class()
Elliptic curves in class 32.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality | CM discriminant |
---|---|---|---|---|---|---|
32.a1 | 32a3 | [0, 0, 0, -11, -14] | [2] | 4 | -16 | |
32.a2 | 32a4 | [0, 0, 0, -11, 14] | [4] | 4 | -16 | |
32.a3 | 32a2 | [0, 0, 0, -1, 0] | [2, 2] | 2 | -4 | |
32.a4 | 32a1 | [0, 0, 0, 4, 0] | [4] | 1 | \(\Gamma_0(N)\)-optimal | -4 |
Rank
sage: E.rank()
The elliptic curves in class 32.a have rank \(0\).
Complex multiplication
Each elliptic curve in class 32.a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 32.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.