# Properties

 Label 3.2.ac_c_ab Base Field $\F_{2}$ Dimension $3$ Ordinary Yes $p$-rank $3$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2}$ Dimension: $3$ L-polynomial: $1 - 2 x + 2 x^{2} - x^{3} + 4 x^{4} - 8 x^{5} + 8 x^{6}$ Frobenius angles: $\pm0.161334789180$, $\pm0.327009058845$, $\pm0.739882802642$ Angle rank: $3$ (numerical) Number field: 6.0.2464727.1 Galois group: $S_4\times C_2$ Jacobians: 2

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $3$ Slopes: $[0, 0, 0, 1, 1, 1]$

## Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

• $y^2+(x^4+x^2+x+1)y=x^8+x^7+x^5+x^3+x^2+1$
• $x^4+x^3y+x^2z^2+xz^3+y^4+y^3z+y^2z^2=0$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4 104 592 11024 31604 246272 2553436 16778528 154949488 1118339144

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 1 5 10 33 31 62 155 257 586 1065

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The endomorphism algebra of this simple isogeny class is 6.0.2464727.1.
All geometric endomorphisms are defined over $\F_{2}$.

## 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 3.2.c_c_b $2$ 3.4.a_i_ab