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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 86640.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
86640.d1 | 86640bq2 | \([0, -1, 0, -21059416, -37190773520]\) | \(781484460931/900\) | \(1189555929092505600\) | \([2]\) | \(3502080\) | \(2.7524\) | |
86640.d2 | 86640bq1 | \([0, -1, 0, -1305496, -590710544]\) | \(-186169411/6480\) | \(-8564802689466040320\) | \([2]\) | \(1751040\) | \(2.4058\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 86640.d have rank \(0\).
Complex multiplication
The elliptic curves in class 86640.d do not have complex multiplication.Modular form 86640.2.a.d
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 LMFDB 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 LMFDB labels.