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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 83259.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
83259.k1 | 83259f1 | \([1, -1, 0, -3720321, -2757359880]\) | \(13132563308857/20231937\) | \(8773097980363424433\) | \([2]\) | \(1881600\) | \(2.5341\) | \(\Gamma_0(N)\)-optimal |
83259.k2 | 83259f2 | \([1, -1, 0, -2622816, -4417445943]\) | \(-4601630708137/16783438221\) | \(-7277738555641479364989\) | \([2]\) | \(3763200\) | \(2.8807\) |
Rank
sage: E.rank()
The elliptic curves in class 83259.k have rank \(0\).
Complex multiplication
The elliptic curves in class 83259.k do not have complex multiplication.Modular form 83259.2.a.k
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.