# Properties

 Label 2.4.a_ab 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: $( 1 - 3 x + 4 x^{2} )( 1 + 3 x + 4 x^{2} )$ Frobenius angles: $\pm0.230053456163$, $\pm0.769946543837$ Angle rank: $1$ (numerical) Jacobians: 2

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 2 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 16 256 4144 82944 1047376 17172736 268460656 4236447744 68719003024 1096996485376

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 15 65 319 1025 4191 16385 64639 262145 1046175

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The isogeny class factors as 1.4.ad $\times$ 1.4.d and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2^{2}}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$
All geometric endomorphisms are defined over $\F_{2^{4}}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.ag_r $2$ 2.16.ac_bh 2.4.g_r $2$ 2.16.ac_bh 2.4.a_b $4$ 2.256.ck_cer
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.ag_r $2$ 2.16.ac_bh 2.4.g_r $2$ 2.16.ac_bh 2.4.a_b $4$ 2.256.ck_cer 2.4.ad_f $6$ (not in LMFDB) 2.4.d_f $6$ (not in LMFDB)