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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 4050.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4050.c1 | 4050f4 | \([1, -1, 0, -242367, -45865459]\) | \(-189613868625/128\) | \(-1062882000000\) | \([]\) | \(18144\) | \(1.6234\) | |
4050.c2 | 4050f3 | \([1, -1, 0, -2367, -89459]\) | \(-1159088625/2097152\) | \(-2654208000000\) | \([]\) | \(6048\) | \(1.0741\) | |
4050.c3 | 4050f1 | \([1, -1, 0, -117, 541]\) | \(-140625/8\) | \(-10125000\) | \([]\) | \(864\) | \(0.10113\) | \(\Gamma_0(N)\)-optimal |
4050.c4 | 4050f2 | \([1, -1, 0, 633, 791]\) | \(3375/2\) | \(-16607531250\) | \([]\) | \(2592\) | \(0.65044\) |
Rank
sage: E.rank()
The elliptic curves in class 4050.c have rank \(1\).
Complex multiplication
The elliptic curves in class 4050.c do not have complex multiplication.Modular form 4050.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 21 & 7 \\ 3 & 1 & 7 & 21 \\ 21 & 7 & 1 & 3 \\ 7 & 21 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.