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SageMath
sage: E = EllipticCurve("c1")
sage: E.isogeny_class()
Elliptic curves in class 6776c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
6776.g4 | 6776c1 | [0, 0, 0, 121, -2662] | [2] | 2560 | \(\Gamma_0(N)\)-optimal |
6776.g3 | 6776c2 | [0, 0, 0, -2299, -39930] | [2, 2] | 5120 | |
6776.g1 | 6776c3 | [0, 0, 0, -36179, -2648690] | [2] | 10240 | |
6776.g2 | 6776c4 | [0, 0, 0, -7139, 183678] | [2] | 10240 |
Rank
sage: E.rank()
The elliptic curves in class 6776c have rank \(1\).
Complex multiplication
The elliptic curves in class 6776c do not have complex multiplication.Modular form 6776.2.a.c
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.