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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 129360p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
129360.y1 | 129360p1 | \([0, -1, 0, -307736, 65809920]\) | \(26752959989284/169785\) | \(20454436316160\) | \([2]\) | \(958464\) | \(1.7392\) | \(\Gamma_0(N)\)-optimal |
129360.y2 | 129360p2 | \([0, -1, 0, -301856, 68439456]\) | \(-12624273557282/1067664675\) | \(-257248627402905600\) | \([2]\) | \(1916928\) | \(2.0858\) |
Rank
sage: E.rank()
The elliptic curves in class 129360p have rank \(0\).
Complex multiplication
The elliptic curves in class 129360p do not have complex multiplication.Modular form 129360.2.a.p
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.