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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 406560bt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
406560.bt1 | 406560bt1 | \([0, -1, 0, -165810, 26042700]\) | \(4446542056384/25725\) | \(2916698030400\) | \([2]\) | \(2073600\) | \(1.5821\) | \(\Gamma_0(N)\)-optimal |
406560.bt2 | 406560bt2 | \([0, -1, 0, -162785, 27035505]\) | \(-65743598656/5294205\) | \(-38416413098004480\) | \([2]\) | \(4147200\) | \(1.9287\) |
Rank
sage: E.rank()
The elliptic curves in class 406560bt have rank \(1\).
Complex multiplication
The elliptic curves in class 406560bt do not have complex multiplication.Modular form 406560.2.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.