# Properties

 Label 2.4.ac_f Base Field $\F_{2^{2}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{2}}$ Dimension: $2$ L-polynomial: $( 1 - 3 x + 4 x^{2} )( 1 + x + 4 x^{2} )$ Frobenius angles: $\pm0.230053456163$, $\pm0.580430623255$ Angle rank: $2$ (numerical) Jacobians: 4

This isogeny class is not 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:

• $y^2+(x^2+x+a+1)y=x^5+ax^4+ax^3+ax^2+(a+1)x+1$
• $y^2+(x^2+x+a)y=x^5+ax^4+(a+1)x^3+ax^2+ax+a$
• $y^2+(x^2+x)y=x^5+(a+1)x^4+x^3+(a+1)x$
• $y^2+(x^2+x)y=x^5+(a+1)x^4+x^3+x^2+ax$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 12 384 3996 69120 1175052 17006976 262951932 4280186880 68662073196 1096502123904

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 23 63 271 1143 4151 16047 65311 261927 1045703

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The isogeny class factors as 1.4.ad $\times$ 1.4.b and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{2^{2}}$.

## 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 2.4.ae_l $2$ 2.16.g_z 2.4.c_f $2$ 2.16.g_z 2.4.e_l $2$ 2.16.g_z