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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 259210.bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
259210.bp1 | 259210bp2 | \([1, 0, 0, -630050, -192537500]\) | \(544737993463/20000\) | \(1015526198540000\) | \([2]\) | \(3942400\) | \(1.9671\) | |
259210.bp2 | 259210bp1 | \([1, 0, 0, -37570, -3299388]\) | \(-115501303/25600\) | \(-1299873534131200\) | \([2]\) | \(1971200\) | \(1.6206\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 259210.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 259210.bp do not have complex multiplication.Modular form 259210.2.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 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.