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SageMath
E = EllipticCurve("ed1")
E.isogeny_class()
Elliptic curves in class 206910.ed
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
206910.ed1 | 206910f2 | \([1, -1, 1, -25402037, -49271404201]\) | \(1403607530712116449/39475350\) | \(50981150090064150\) | \([2]\) | \(11468800\) | \(2.7163\) | |
206910.ed2 | 206910f1 | \([1, -1, 1, -1585607, -771626149]\) | \(-341370886042369/1817528220\) | \(-2347279478883585180\) | \([2]\) | \(5734400\) | \(2.3698\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 206910.ed have rank \(1\).
Complex multiplication
The elliptic curves in class 206910.ed do not have complex multiplication.Modular form 206910.2.a.ed
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.