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SageMath
sage: E = EllipticCurve("a1")
sage: E.isogeny_class()
Elliptic curves in class 27.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality | CM discriminant |
---|---|---|---|---|---|---|
27.a1 | 27a2 | [0, 0, 1, -270, -1708] | [] | 3 | -27 | |
27.a2 | 27a4 | [0, 0, 1, -30, 63] | [3] | 9 | -27 | |
27.a3 | 27a1 | [0, 0, 1, 0, -7] | [3] | 1 | \(\Gamma_0(N)\)-optimal | -3 |
27.a4 | 27a3 | [0, 0, 1, 0, 0] | [3] | 3 | -3 |
Rank
sage: E.rank()
The elliptic curves in class 27.a have rank \(0\).
Complex multiplication
Each elliptic curve in class 27.a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 27.2.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.
\(\left(\begin{array}{rrrr} 1 & 27 & 3 & 9 \\ 27 & 1 & 9 & 3 \\ 3 & 9 & 1 & 3 \\ 9 & 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.