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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 53550bk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53550.a2 | 53550bk1 | \([1, -1, 0, -9567, 1312591]\) | \(-8502154921/60184250\) | \(-685536222656250\) | \([]\) | \(380160\) | \(1.5300\) | \(\Gamma_0(N)\)-optimal |
53550.a1 | 53550bk2 | \([1, -1, 0, -1252692, 539968216]\) | \(-19085751483878521/80001320\) | \(-911265035625000\) | \([]\) | \(1140480\) | \(2.0793\) |
Rank
sage: E.rank()
The elliptic curves in class 53550bk have rank \(1\).
Complex multiplication
The elliptic curves in class 53550bk do not have complex multiplication.Modular form 53550.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.