Properties

 Label 489762.dg Number of curves $6$ Conductor $489762$ CM no Rank $2$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("489762.dg1")

sage: E.isogeny_class()

Elliptic curves in class 489762.dg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
489762.dg1 489762dg5 [1, -1, 1, -120362846, 508285978737] [2] 75497472 $$\Gamma_0(N)$$-optimal*
489762.dg2 489762dg4 [1, -1, 1, -26943026, -53820986943] [2] 37748736
489762.dg3 489762dg3 [1, -1, 1, -7717586, 7510210881] [2, 2] 37748736 $$\Gamma_0(N)$$-optimal*
489762.dg4 489762dg2 [1, -1, 1, -1755266, -765489279] [2, 2] 18874368 $$\Gamma_0(N)$$-optimal*
489762.dg5 489762dg1 [1, -1, 1, 191614, -66169983] [2] 9437184 $$\Gamma_0(N)$$-optimal*
489762.dg6 489762dg6 [1, -1, 1, 9530554, 36307705425] [2] 75497472
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 489762.dg5.

Rank

sage: E.rank()

The elliptic curves in class 489762.dg have rank $$2$$.

Modular form 489762.2.a.dg

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - 2q^{5} - q^{7} + q^{8} - 2q^{10} - 4q^{11} - q^{14} + q^{16} + 6q^{17} - 4q^{19} + O(q^{20})$$

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.

$$\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with LMFDB labels.