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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 311542bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
311542.bp2 | 311542bp1 | \([1, 1, 0, -205629, 37024877]\) | \(-338608873/13552\) | \(-38484439110768112\) | \([2]\) | \(3981312\) | \(1.9507\) | \(\Gamma_0(N)\)-optimal |
311542.bp1 | 311542bp2 | \([1, 1, 0, -3321049, 2328104745]\) | \(1426487591593/2156\) | \(6122524403985836\) | \([2]\) | \(7962624\) | \(2.2972\) |
Rank
sage: E.rank()
The elliptic curves in class 311542bp have rank \(0\).
Complex multiplication
The elliptic curves in class 311542bp do not have complex multiplication.Modular form 311542.2.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.