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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 162450.ek
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
162450.ek1 | 162450bl2 | \([1, -1, 1, -5800730, -5375240103]\) | \(276288773643091/41990400\) | \(3280638437100000000\) | \([2]\) | \(3932160\) | \(2.5654\) | |
162450.ek2 | 162450bl1 | \([1, -1, 1, -328730, -100232103]\) | \(-50284268371/26542080\) | \(-2073687505920000000\) | \([2]\) | \(1966080\) | \(2.2188\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 162450.ek have rank \(1\).
Complex multiplication
The elliptic curves in class 162450.ek do not have complex multiplication.Modular form 162450.2.a.ek
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.