# Properties

 Label 493680.eg Number of curves $2$ Conductor $493680$ CM no Rank $2$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("eg1")

sage: E.isogeny_class()

## Elliptic curves in class 493680.eg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
493680.eg1 493680eg2 [0, 1, 0, -277856, 35917044]  5898240 $$\Gamma_0(N)$$-optimal*
493680.eg2 493680eg1 [0, 1, 0, 51264, 3926580]  2949120 $$\Gamma_0(N)$$-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 493680.eg1.

## Rank

sage: E.rank()

The elliptic curves in class 493680.eg have rank $$2$$.

## Complex multiplication

The elliptic curves in class 493680.eg do not have complex multiplication.

## Modular form 493680.2.a.eg

sage: E.q_eigenform(10)

$$q + q^{3} - q^{5} - 2q^{7} + q^{9} - q^{15} - q^{17} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 