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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 53550bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53550.s2 | 53550bc1 | \([1, -1, 0, -4842, -180684]\) | \(-1102302937/616896\) | \(-7026831000000\) | \([2]\) | \(98304\) | \(1.1684\) | \(\Gamma_0(N)\)-optimal |
53550.s1 | 53550bc2 | \([1, -1, 0, -85842, -9657684]\) | \(6141556990297/1019592\) | \(11613790125000\) | \([2]\) | \(196608\) | \(1.5149\) |
Rank
sage: E.rank()
The elliptic curves in class 53550bc have rank \(1\).
Complex multiplication
The elliptic curves in class 53550bc do not have complex multiplication.Modular form 53550.2.a.bc
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.