# Properties

 Label 2.19.al_ci 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: $1 - 11 x + 60 x^{2} - 209 x^{3} + 361 x^{4}$ Frobenius angles: $\pm0.0899168499442$, $\pm0.402539378619$ Angle rank: $2$ (numerical) Number field: 4.0.444312.2 Galois group: $D_{4}$ Jacobians: 4

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 4 curves, and hence is principally polarizable:

• $y^2=8x^6+5x^4+4x^3+15x^2+x+8$
• $y^2=12x^6+9x^5+5x^4+x^3+x^2+17x+12$
• $y^2=14x^6+5x^5+18x^4+11x^3+18x^2+3x+6$
• $y^2=16x^6+9x^5+10x^4+12x^3+9x^2+8x+8$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 202 129684 47196088 16911831072 6122057290822 2213225055301824 799072753925610646 288447636123620162688 104127515019253716936856 37589972043459737592720084

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 9 361 6882 129769 2472459 47043970 893945649 16983929425 322688208246 6131066027161

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The endomorphism algebra of this simple isogeny class is 4.0.444312.2.
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.l_ci $2$ (not in LMFDB)