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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 32851k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32851.f2 | 32851k1 | \([0, -1, 1, -2647, -51640]\) | \(-43614208/91\) | \(-4281175171\) | \([]\) | \(25272\) | \(0.73371\) | \(\Gamma_0(N)\)-optimal |
32851.f3 | 32851k2 | \([0, -1, 1, 4573, -262103]\) | \(224755712/753571\) | \(-35452411591051\) | \([]\) | \(75816\) | \(1.2830\) | |
32851.f1 | 32851k3 | \([0, -1, 1, -42357, 8283850]\) | \(-178643795968/524596891\) | \(-24680122906955971\) | \([]\) | \(227448\) | \(1.8323\) |
Rank
sage: E.rank()
The elliptic curves in class 32851k have rank \(1\).
Complex multiplication
The elliptic curves in class 32851k do not have complex multiplication.Modular form 32851.2.a.k
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.