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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 1710i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1710.k2 | 1710i1 | \([1, -1, 0, -13104, 583308]\) | \(-341370886042369/1817528220\) | \(-1324978072380\) | \([2]\) | \(4480\) | \(1.1708\) | \(\Gamma_0(N)\)-optimal |
1710.k1 | 1710i2 | \([1, -1, 0, -209934, 37075590]\) | \(1403607530712116449/39475350\) | \(28777530150\) | \([2]\) | \(8960\) | \(1.5174\) |
Rank
sage: E.rank()
The elliptic curves in class 1710i have rank \(0\).
Complex multiplication
The elliptic curves in class 1710i do not have complex multiplication.Modular form 1710.2.a.i
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 Cremona 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 Cremona labels.