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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 68970.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68970.s1 | 68970y2 | \([1, 0, 1, -2822449, 1824866822]\) | \(1403607530712116449/39475350\) | \(69932990521350\) | \([2]\) | \(1433600\) | \(2.1670\) | |
68970.s2 | 68970y1 | \([1, 0, 1, -176179, 28578746]\) | \(-341370886042369/1817528220\) | \(-3219862110951420\) | \([2]\) | \(716800\) | \(1.8204\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 68970.s have rank \(1\).
Complex multiplication
The elliptic curves in class 68970.s do not have complex multiplication.Modular form 68970.2.a.s
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.