Properties

 Label 66990.bg 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 66990.bg

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

Rank

sage: E.rank()

The elliptic curves in class 66990.bg have rank $$0$$.

Modular form None

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.