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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 66.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
66.c1 | 66c3 | \([1, 0, 0, -10065, -389499]\) | \(112763292123580561/1932612\) | \(1932612\) | \([2]\) | \(100\) | \(0.74785\) | |
66.c2 | 66c4 | \([1, 0, 0, -10055, -390309]\) | \(-112427521449300721/466873642818\) | \(-466873642818\) | \([2]\) | \(200\) | \(1.0944\) | |
66.c3 | 66c1 | \([1, 0, 0, -45, 81]\) | \(10091699281/2737152\) | \(2737152\) | \([10]\) | \(20\) | \(-0.056866\) | \(\Gamma_0(N)\)-optimal |
66.c4 | 66c2 | \([1, 0, 0, 115, 561]\) | \(168105213359/228637728\) | \(-228637728\) | \([10]\) | \(40\) | \(0.28971\) |
Rank
sage: E.rank()
The elliptic curves in class 66.c have rank \(0\).
Complex multiplication
The elliptic curves in class 66.c do not have complex multiplication.Modular form 66.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.