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SageMath
E = EllipticCurve("et1")
E.isogeny_class()
Elliptic curves in class 162450et
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
162450.z2 | 162450et1 | \([1, -1, 0, -28767, -180359]\) | \(132651/76\) | \(1508408559562500\) | \([2]\) | \(737280\) | \(1.6022\) | \(\Gamma_0(N)\)-optimal |
162450.z1 | 162450et2 | \([1, -1, 0, -299517, 62904391]\) | \(149721291/722\) | \(14329881315843750\) | \([2]\) | \(1474560\) | \(1.9488\) |
Rank
sage: E.rank()
The elliptic curves in class 162450et have rank \(2\).
Complex multiplication
The elliptic curves in class 162450et do not have complex multiplication.Modular form 162450.2.a.et
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.