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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 574.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
574.c1 | 574f2 | \([1, 0, 1, -2335, -43598]\) | \(1407074115849193/460816384\) | \(460816384\) | \([]\) | \(360\) | \(0.63599\) | |
574.c2 | 574f1 | \([1, 0, 1, -80, 190]\) | \(55611739513/15438304\) | \(15438304\) | \([3]\) | \(120\) | \(0.086689\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 574.c have rank \(1\).
Complex multiplication
The elliptic curves in class 574.c do not have complex multiplication.Modular form 574.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.