# Properties

 Label 84966cw Number of curves $4$ Conductor $84966$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("cw1")

sage: E.isogeny_class()

## Elliptic curves in class 84966cw

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
84966.ct4 84966cw1 $$[1, 1, 1, 13866, -169807149]$$ $$103823/4386816$$ $$-12457508356120375296$$ $$[4]$$ $$2654208$$ $$2.3430$$ $$\Gamma_0(N)$$-optimal
84966.ct3 84966cw2 $$[1, 1, 1, -4517654, -3631888429]$$ $$3590714269297/73410624$$ $$208468616396951905344$$ $$[2, 2]$$ $$5308416$$ $$2.6895$$
84966.ct2 84966cw3 $$[1, 1, 1, -9615614, 6062392307]$$ $$34623662831857/14438442312$$ $$41001723288850499049672$$ $$[2]$$ $$10616832$$ $$3.0361$$
84966.ct1 84966cw4 $$[1, 1, 1, -71924014, -234808740685]$$ $$14489843500598257/6246072$$ $$17737350764866706232$$ $$[2]$$ $$10616832$$ $$3.0361$$

## Rank

sage: E.rank()

The elliptic curves in class 84966cw have rank $$1$$.

## Complex multiplication

The elliptic curves in class 84966cw do not have complex multiplication.

## Modular form 84966.2.a.cw

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + q^{8} + q^{9} - 2q^{10} - q^{12} + 6q^{13} + 2q^{15} + q^{16} + q^{18} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.