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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 12642z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12642.y2 | 12642z1 | \([1, 1, 1, 7790, -588001]\) | \(444369620591/1540767744\) | \(-181269784313856\) | \([]\) | \(63504\) | \(1.4189\) | \(\Gamma_0(N)\)-optimal |
12642.y1 | 12642z2 | \([1, 1, 1, -2935150, 1934508239]\) | \(-23769846831649063249/3261823333284\) | \(-383750253337529316\) | \([]\) | \(444528\) | \(2.3919\) |
Rank
sage: E.rank()
The elliptic curves in class 12642z have rank \(0\).
Complex multiplication
The elliptic curves in class 12642z do not have complex multiplication.Modular form 12642.2.a.z
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.