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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 119952bk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
119952.bp4 | 119952bk1 | \([0, 0, 0, -311630151, -2137094698970]\) | \(-152435594466395827792/1646846627220711\) | \(-36158373657255268692999936\) | \([2]\) | \(26542080\) | \(3.7201\) | \(\Gamma_0(N)\)-optimal |
119952.bp3 | 119952bk2 | \([0, 0, 0, -4998939771, -136039468613510]\) | \(157304700372188331121828/18069292138401\) | \(1586926690228683686421504\) | \([2, 2]\) | \(53084160\) | \(4.0667\) | |
119952.bp2 | 119952bk3 | \([0, 0, 0, -5011799331, -135304372441406]\) | \(79260902459030376659234/842751810121431609\) | \(148028526018239955088831875072\) | \([2]\) | \(106168320\) | \(4.4133\) | |
119952.bp1 | 119952bk4 | \([0, 0, 0, -79983034131, -8706526495316174]\) | \(322159999717985454060440834/4250799\) | \(746648660747630592\) | \([2]\) | \(106168320\) | \(4.4133\) |
Rank
sage: E.rank()
The elliptic curves in class 119952bk have rank \(0\).
Complex multiplication
The elliptic curves in class 119952bk do not have complex multiplication.Modular form 119952.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.