# Properties

 Label 2.19.ak_bz 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: $1 - 10 x + 51 x^{2} - 190 x^{3} + 361 x^{4}$ Frobenius angles: $\pm0.0769799514810$, $\pm0.443626041964$ Angle rank: $2$ (numerical) Number field: 4.0.46224.1 Galois group: $D_{4}$ Jacobians: 6

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 6 curves, and hence is principally polarizable:

• $y^2=16x^6+7x^4+12x^3+x^2+8x+4$
• $y^2=8x^6+6x^5+13x^4+11x^3+14x^2+3x+12$
• $y^2=6x^6+8x^5+13x^4+4x^3+6x^2+14x+16$
• $y^2=14x^6+x^5+16x^4+8x^3+11x^2+13x+3$
• $y^2=13x^6+3x^5+15x^4+18x^3+11x^2+16x+13$
• $y^2=6x^6+x^5+16x^4+17x^3+x^2+15x+18$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 213 130569 46774800 16859199849 6122283767853 2213551925280000 799063743815563173 288442310363975783049 104127192529352677789200 37589990180663150170659609

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 10 364 6820 129364 2472550 47050918 893935570 16983615844 322687208860 6131068985404

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The endomorphism algebra of this simple isogeny class is 4.0.46224.1.
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_bz $2$ (not in LMFDB)