# Properties

 Label 2.19.ak_ce Base Field $\F_{19}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{19}$ Dimension: $2$ L-polynomial: $1 - 10 x + 56 x^{2} - 190 x^{3} + 361 x^{4}$ Frobenius angles: $\pm0.159522866725$, $\pm0.412959486295$ Angle rank: $2$ (numerical) Number field: 4.0.969024.4 Galois group: $D_{4}$ Jacobians: 8

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## Point counts

This isogeny class contains the Jacobians of 8 curves, and hence is principally polarizable:

• $y^2=13x^6+7x^5+4x^4+5x^3+17x^2+13x+1$
• $y^2=6x^6+14x^5+x^4+13x^3+8x^2+5x+16$
• $y^2=12x^6+2x^5+12x^4+9x^3+16x^2+6x+12$
• $y^2=x^6+9x^5+16x^4+7x^3+4x^2+16x+4$
• $y^2=18x^6+17x^5+x^4+5x^3+5x+8$
• $y^2=8x^6+x^5+6x^4+3x^3+15x^2+3x+12$
• $y^2=3x^6+3x^5+12x^4+17x^3+15x^2+15$
• $y^2=7x^6+10x^5+16x^4+17x^3+x^2+18x+12$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 218 134724 47812850 16980074064 6129701550578 2213909471902500 799106920552060058 288446836342806918144 104127193159165489632650 37589936628568662296514084

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 10 374 6970 130294 2475550 47058518 893983870 16983882334 322687210810 6131060250854

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The endomorphism algebra of this simple isogeny class is 4.0.969024.4.
All geometric endomorphisms are defined over $\F_{19}$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.19.k_ce $2$ (not in LMFDB)