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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 27225bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27225.g2 | 27225bm1 | \([1, -1, 1, -6755, 215372]\) | \(-24729001\) | \(-1378265625\) | \([]\) | \(20160\) | \(0.83606\) | \(\Gamma_0(N)\)-optimal |
27225.g1 | 27225bm2 | \([1, -1, 1, -68630, -27034378]\) | \(-121\) | \(-295443477095765625\) | \([]\) | \(221760\) | \(2.0350\) |
Rank
sage: E.rank()
The elliptic curves in class 27225bm have rank \(1\).
Complex multiplication
The elliptic curves in class 27225bm do not have complex multiplication.Modular form 27225.2.a.bm
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.
\(\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.