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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 82810bk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
82810.bl2 | 82810bk1 | \([1, 1, 0, -9481917, -9137546179]\) | \(166021325905681/32614400000\) | \(18520714933765990400000\) | \([2]\) | \(9031680\) | \(2.9894\) | \(\Gamma_0(N)\)-optimal |
82810.bl1 | 82810bk2 | \([1, 1, 0, -46580797, 114127192509]\) | \(19683218700810001/1478750000000\) | \(839736656455628750000000\) | \([2]\) | \(18063360\) | \(3.3359\) |
Rank
sage: E.rank()
The elliptic curves in class 82810bk have rank \(1\).
Complex multiplication
The elliptic curves in class 82810bk do not have complex multiplication.Modular form 82810.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.