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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 6720.cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6720.cd1 | 6720cf3 | \([0, 1, 0, -2625, -52545]\) | \(15267472418/36015\) | \(4720558080\) | \([2]\) | \(4096\) | \(0.73620\) | |
6720.cd2 | 6720cf2 | \([0, 1, 0, -225, -225]\) | \(19307236/11025\) | \(722534400\) | \([2, 2]\) | \(2048\) | \(0.38963\) | |
6720.cd3 | 6720cf1 | \([0, 1, 0, -145, 623]\) | \(20720464/105\) | \(1720320\) | \([2]\) | \(1024\) | \(0.043052\) | \(\Gamma_0(N)\)-optimal |
6720.cd4 | 6720cf4 | \([0, 1, 0, 895, -897]\) | \(604223422/354375\) | \(-46448640000\) | \([4]\) | \(4096\) | \(0.73620\) |
Rank
sage: E.rank()
The elliptic curves in class 6720.cd have rank \(0\).
Complex multiplication
The elliptic curves in class 6720.cd do not have complex multiplication.Modular form 6720.2.a.cd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.