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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 162450.ck
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
162450.ck1 | 162450ez2 | \([1, -1, 0, -43998567, 111262077341]\) | \(651038076963/7220000\) | \(104464834792500937500000\) | \([2]\) | \(33177600\) | \(3.2313\) | |
162450.ck2 | 162450ez1 | \([1, -1, 0, -5010567, -1530206659]\) | \(961504803/486400\) | \(7037630975494800000000\) | \([2]\) | \(16588800\) | \(2.8847\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 162450.ck have rank \(0\).
Complex multiplication
The elliptic curves in class 162450.ck do not have complex multiplication.Modular form 162450.2.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 LMFDB 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 LMFDB labels.