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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 660c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
660.c3 | 660c1 | \([0, 1, 0, -41, 120]\) | \(-488095744/200475\) | \(-3207600\) | \([6]\) | \(144\) | \(-0.043272\) | \(\Gamma_0(N)\)-optimal |
660.c2 | 660c2 | \([0, 1, 0, -716, 7140]\) | \(158792223184/16335\) | \(4181760\) | \([6]\) | \(288\) | \(0.30330\) | |
660.c4 | 660c3 | \([0, 1, 0, 319, -1356]\) | \(223673040896/187171875\) | \(-2994750000\) | \([2]\) | \(432\) | \(0.50603\) | |
660.c1 | 660c4 | \([0, 1, 0, -1556, -13356]\) | \(1628514404944/664335375\) | \(170069856000\) | \([2]\) | \(864\) | \(0.85261\) |
Rank
sage: E.rank()
The elliptic curves in class 660c have rank \(1\).
Complex multiplication
The elliptic curves in class 660c do not have complex multiplication.Modular form 660.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.