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SageMath
sage: E = EllipticCurve("cc1")
sage: E.isogeny_class()
Elliptic curves in class 6720.cc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
6720.cc1 | 6720cg7 | [0, 1, 0, -412865, -64377537] | [2] | 110592 | |
6720.cc2 | 6720cg4 | [0, 1, 0, -368705, -86295105] | [2] | 36864 | |
6720.cc3 | 6720cg6 | [0, 1, 0, -172865, 26870463] | [2, 2] | 55296 | |
6720.cc4 | 6720cg3 | [0, 1, 0, -171585, 27299775] | [2] | 27648 | |
6720.cc5 | 6720cg2 | [0, 1, 0, -23105, -1346625] | [2, 2] | 18432 | |
6720.cc6 | 6720cg5 | [0, 1, 0, -5185, -3364417] | [4] | 36864 | |
6720.cc7 | 6720cg1 | [0, 1, 0, -2625, 17343] | [2] | 9216 | \(\Gamma_0(N)\)-optimal |
6720.cc8 | 6720cg8 | [0, 1, 0, 46655, 90662975] | [4] | 110592 |
Rank
sage: E.rank()
The elliptic curves in class 6720.cc have rank \(0\).
Complex multiplication
The elliptic curves in class 6720.cc do not have complex multiplication.Modular form 6720.2.a.cc
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.