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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 7605.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
7605.r1 | 7605g2 | \([1, -1, 0, -31719, 2084850]\) | \(12326391/625\) | \(178950926919375\) | \([2]\) | \(29952\) | \(1.4931\) | |
7605.r2 | 7605g1 | \([1, -1, 0, 1236, 127323]\) | \(729/25\) | \(-7158037076775\) | \([2]\) | \(14976\) | \(1.1466\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 7605.r have rank \(1\).
Complex multiplication
The elliptic curves in class 7605.r do not have complex multiplication.Modular form 7605.2.a.r
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.