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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 453024bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
453024.bp2 | 453024bp1 | \([0, 0, 0, -19965, -1075448]\) | \(10648000/117\) | \(9670512151872\) | \([2]\) | \(737280\) | \(1.3064\) | \(\Gamma_0(N)\)-optimal* |
453024.bp1 | 453024bp2 | \([0, 0, 0, -36300, 937024]\) | \(1000000/507\) | \(2681955370119168\) | \([2]\) | \(1474560\) | \(1.6529\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 453024bp have rank \(2\).
Complex multiplication
The elliptic curves in class 453024bp do not have complex multiplication.Modular form 453024.2.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.