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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 133200ck
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
133200.cw3 | 133200ck1 | \([0, 0, 0, -12000, 502000]\) | \(4096000/37\) | \(1726272000000\) | \([]\) | \(207360\) | \(1.1706\) | \(\Gamma_0(N)\)-optimal |
133200.cw2 | 133200ck2 | \([0, 0, 0, -84000, -9074000]\) | \(1404928000/50653\) | \(2363266368000000\) | \([]\) | \(622080\) | \(1.7199\) | |
133200.cw1 | 133200ck3 | \([0, 0, 0, -6744000, -6741002000]\) | \(727057727488000/37\) | \(1726272000000\) | \([]\) | \(1866240\) | \(2.2693\) |
Rank
sage: E.rank()
The elliptic curves in class 133200ck have rank \(1\).
Complex multiplication
The elliptic curves in class 133200ck do not have complex multiplication.Modular form 133200.2.a.ck
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.