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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 8002.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8002.c1 | 8002c2 | \([1, 1, 1, -699134, -225294533]\) | \(37792447651021184768737/2098200707072\) | \(2098200707072\) | \([2]\) | \(72624\) | \(1.8323\) | |
8002.c2 | 8002c1 | \([1, 1, 1, -43774, -3520709]\) | \(9276262566809508577/68736656605184\) | \(68736656605184\) | \([2]\) | \(36312\) | \(1.4858\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8002.c have rank \(1\).
Complex multiplication
The elliptic curves in class 8002.c do not have complex multiplication.Modular form 8002.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.