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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 18240.cp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
18240.cp1 | 18240bn3 | \([0, 1, 0, -39745, -3058657]\) | \(26487576322129/44531250\) | \(11673600000000\) | \([2]\) | \(49152\) | \(1.4026\) | |
18240.cp2 | 18240bn2 | \([0, 1, 0, -3265, -16225]\) | \(14688124849/8122500\) | \(2129264640000\) | \([2, 2]\) | \(24576\) | \(1.0561\) | |
18240.cp3 | 18240bn1 | \([0, 1, 0, -1985, 33183]\) | \(3301293169/22800\) | \(5976883200\) | \([2]\) | \(12288\) | \(0.70949\) | \(\Gamma_0(N)\)-optimal |
18240.cp4 | 18240bn4 | \([0, 1, 0, 12735, -115425]\) | \(871257511151/527800050\) | \(-138359616307200\) | \([4]\) | \(49152\) | \(1.4026\) |
Rank
sage: E.rank()
The elliptic curves in class 18240.cp have rank \(0\).
Complex multiplication
The elliptic curves in class 18240.cp do not have complex multiplication.Modular form 18240.2.a.cp
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.