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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 25872bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25872.e2 | 25872bp1 | \([0, -1, 0, -50864, -4901952]\) | \(-10358806345399/1445216256\) | \(-2030424784109568\) | \([2]\) | \(129024\) | \(1.6687\) | \(\Gamma_0(N)\)-optimal |
25872.e1 | 25872bp2 | \([0, -1, 0, -839344, -295693376]\) | \(46546832455691959/748268928\) | \(1051263968477184\) | \([2]\) | \(258048\) | \(2.0153\) |
Rank
sage: E.rank()
The elliptic curves in class 25872bp have rank \(1\).
Complex multiplication
The elliptic curves in class 25872bp do not have complex multiplication.Modular form 25872.2.a.bp
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.