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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 51842.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51842.p1 | 51842o2 | \([1, 1, 1, -4510794, 3681045815]\) | \(582810602977/829472\) | \(14446311880284610592\) | \([2]\) | \(2027520\) | \(2.5794\) | |
51842.p2 | 51842o1 | \([1, 1, 1, -363434, 21415351]\) | \(304821217/164864\) | \(2871316647013090304\) | \([2]\) | \(1013760\) | \(2.2328\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 51842.p have rank \(0\).
Complex multiplication
The elliptic curves in class 51842.p do not have complex multiplication.Modular form 51842.2.a.p
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.