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SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 54720cc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54720.ea4 | 54720cc1 | \([0, 0, 0, 708, -12656]\) | \(3286064/7695\) | \(-91908587520\) | \([2]\) | \(49152\) | \(0.78768\) | \(\Gamma_0(N)\)-optimal |
54720.ea3 | 54720cc2 | \([0, 0, 0, -5772, -139664]\) | \(445138564/81225\) | \(3880584806400\) | \([2, 2]\) | \(98304\) | \(1.1342\) | |
54720.ea2 | 54720cc3 | \([0, 0, 0, -27372, 1614256]\) | \(23735908082/1954815\) | \(186785482014720\) | \([4]\) | \(196608\) | \(1.4808\) | |
54720.ea1 | 54720cc4 | \([0, 0, 0, -87852, -10022096]\) | \(784767874322/35625\) | \(3404021760000\) | \([2]\) | \(196608\) | \(1.4808\) |
Rank
sage: E.rank()
The elliptic curves in class 54720cc have rank \(0\).
Complex multiplication
The elliptic curves in class 54720cc do not have complex multiplication.Modular form 54720.2.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 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.