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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 66066.bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
66066.bg1 | 66066w4 | \([1, 0, 1, -50460, -4366892]\) | \(8020417344913/187278\) | \(331774400958\) | \([2]\) | \(245760\) | \(1.3229\) | |
66066.bg2 | 66066w2 | \([1, 0, 1, -3270, -63164]\) | \(2181825073/298116\) | \(528130679076\) | \([2, 2]\) | \(122880\) | \(0.97635\) | |
66066.bg3 | 66066w1 | \([1, 0, 1, -850, 8468]\) | \(38272753/4368\) | \(7738178448\) | \([2]\) | \(61440\) | \(0.62977\) | \(\Gamma_0(N)\)-optimal |
66066.bg4 | 66066w3 | \([1, 0, 1, 5200, -334204]\) | \(8780064047/32388174\) | \(-57377625919614\) | \([2]\) | \(245760\) | \(1.3229\) |
Rank
sage: E.rank()
The elliptic curves in class 66066.bg have rank \(0\).
Complex multiplication
The elliptic curves in class 66066.bg do not have complex multiplication.Modular form 66066.2.a.bg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.