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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 32490bt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32490.bq2 | 32490bt1 | \([1, -1, 1, -4746857, 5505632169]\) | \(-50284268371/26542080\) | \(-6243741160620743393280\) | \([2]\) | \(1556480\) | \(2.8863\) | \(\Gamma_0(N)\)-optimal |
32490.bq1 | 32490bt2 | \([1, -1, 1, -83762537, 295050689961]\) | \(276288773643091/41990400\) | \(9877793633013285446400\) | \([2]\) | \(3112960\) | \(3.2329\) |
Rank
sage: E.rank()
The elliptic curves in class 32490bt have rank \(0\).
Complex multiplication
The elliptic curves in class 32490bt do not have complex multiplication.Modular form 32490.2.a.bt
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.