# Properties

 Label 481650.gq Number of curves $4$ Conductor $481650$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("481650.gq1")

sage: E.isogeny_class()

## Elliptic curves in class 481650.gq

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
481650.gq1 481650gq3 [1, 1, 1, -12844088, 17712172031]  21233664 $$\Gamma_0(N)$$-optimal*
481650.gq2 481650gq4 [1, 1, 1, -929588, 183154031]  21233664
481650.gq3 481650gq2 [1, 1, 1, -802838, 276442031] [2, 2] 10616832 $$\Gamma_0(N)$$-optimal*
481650.gq4 481650gq1 [1, 1, 1, -42338, 5704031]  5308416 $$\Gamma_0(N)$$-optimal*
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 481650.gq4.

## Rank

sage: E.rank()

The elliptic curves in class 481650.gq have rank $$1$$.

## Modular form 481650.2.a.gq

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{6} + 4q^{7} + q^{8} + q^{9} + 4q^{11} - q^{12} + 4q^{14} + q^{16} + 2q^{17} + q^{18} + q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 