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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 212160.bk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
212160.bk1 | 212160hq1 | \([0, -1, 0, -42076801, 105067926145]\) | \(31427652507069423952801/654426190080\) | \(171553899172331520\) | \([2]\) | \(9338880\) | \(2.8363\) | \(\Gamma_0(N)\)-optimal |
212160.bk2 | 212160hq2 | \([0, -1, 0, -42030721, 105309486721]\) | \(-31324512477868037557921/143427974919699600\) | \(-37598783057349731942400\) | \([2]\) | \(18677760\) | \(3.1829\) |
Rank
sage: E.rank()
The elliptic curves in class 212160.bk have rank \(2\).
Complex multiplication
The elliptic curves in class 212160.bk do not have complex multiplication.Modular form 212160.2.a.bk
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.