# Properties

 Label 489762bl Number of curves $2$ Conductor $489762$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bl1")

sage: E.isogeny_class()

## Elliptic curves in class 489762bl

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
489762.bl2 489762bl1 $$[1, -1, 0, 52443, -21757163]$$ $$4533086375/60669952$$ $$-213482015080169472$$ $$$$ $$6322176$$ $$2.0060$$ $$\Gamma_0(N)$$-optimal*
489762.bl1 489762bl2 $$[1, -1, 0, -920997, -318266987]$$ $$24553362849625/1755162752$$ $$6175967983139590272$$ $$$$ $$12644352$$ $$2.3525$$ $$\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 489762bl1.

## Rank

sage: E.rank()

The elliptic curves in class 489762bl have rank $$1$$.

## Complex multiplication

The elliptic curves in class 489762bl do not have complex multiplication.

## Modular form 489762.2.a.bl

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 