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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 15870bk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
15870.bm2 | 15870bk1 | \([1, 0, 0, 70875, 7490625]\) | \(265971760991/317400000\) | \(-46986591168600000\) | \([2]\) | \(126720\) | \(1.8853\) | \(\Gamma_0(N)\)-optimal |
15870.bm1 | 15870bk2 | \([1, 0, 0, -415805, 71440377]\) | \(53706380371489/16171875000\) | \(2394017892421875000\) | \([2]\) | \(253440\) | \(2.2318\) |
Rank
sage: E.rank()
The elliptic curves in class 15870bk have rank \(1\).
Complex multiplication
The elliptic curves in class 15870bk do not have complex multiplication.Modular form 15870.2.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.