Properties

Label 66990bf
Number of curves 8
Conductor 66990
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("66990.bg1")
sage: E.isogeny_class()

Elliptic curves in class 66990bf

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
66990.bg7 66990bf1 [1, 0, 1, -7604139, -7854138938] 6 4866048 \(\Gamma_0(N)\)-optimal
66990.bg6 66990bf2 [1, 0, 1, -18232959, 18943242046] 12 9732096  
66990.bg3 66990bf3 [1, 0, 1, -611314554, -5817669148244] 2 14598144  
66990.bg8 66990bf4 [1, 0, 1, 54280671, 132151521202] 6 19464192  
66990.bg5 66990bf5 [1, 0, 1, -260807709, 1620809861146] 6 19464192  
66990.bg2 66990bf6 [1, 0, 1, -611340474, -5817151142228] 4 29196288  
66990.bg4 66990bf7 [1, 0, 1, -549880194, -7033032153524] 2 58392576  
66990.bg1 66990bf8 [1, 0, 1, -673215474, -4568117642228] 2 58392576  

Rank

sage: E.rank()

The elliptic curves in class 66990bf have rank \(0\).

Modular form 66990.2.a.bg

sage: E.q_eigenform(10)
\( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + q^{10} + q^{11} + q^{12} + 2q^{13} - q^{14} - q^{15} + q^{16} - 6q^{17} - q^{18} + 8q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with Cremona labels.