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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 85008bk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
85008.u2 | 85008bk1 | \([0, -1, 0, -62048, -5935104]\) | \(-6449916994998625/8532911772\) | \(-34950806618112\) | \([2]\) | \(258048\) | \(1.5052\) | \(\Gamma_0(N)\)-optimal |
85008.u1 | 85008bk2 | \([0, -1, 0, -993088, -380585600]\) | \(26444015547214434625/46191222\) | \(189199245312\) | \([2]\) | \(516096\) | \(1.8517\) |
Rank
sage: E.rank()
The elliptic curves in class 85008bk have rank \(0\).
Complex multiplication
The elliptic curves in class 85008bk do not have complex multiplication.Modular form 85008.2.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.