# Properties

 Label 466578et Number of curves $2$ Conductor $466578$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("et1")

sage: E.isogeny_class()

## Elliptic curves in class 466578et

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
466578.et2 466578et1 $$[1, -1, 1, 8043610, -41305625035]$$ $$4533086375/60669952$$ $$-770293859527495762034688$$ $$$$ $$68124672$$ $$3.2642$$ $$\Gamma_0(N)$$-optimal*
466578.et1 466578et2 $$[1, -1, 1, -141261350, -604961710027]$$ $$24553362849625/1755162752$$ $$22284360639299350053237888$$ $$$$ $$136249344$$ $$3.6108$$ $$\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 466578et1.

## Rank

sage: E.rank()

The elliptic curves in class 466578et have rank $$1$$.

## Complex multiplication

The elliptic curves in class 466578et do not have complex multiplication.

## Modular form 466578.2.a.et

sage: E.q_eigenform(10)

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