# Properties

 Label 2.4.a_d 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: $1 + 3 x^{2} + 16 x^{4}$ Frobenius angles: $\pm0.311178646770$, $\pm0.688821353230$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{5}, \sqrt{-11})$$ Galois group: $C_2^2$ Jacobians: 6

This isogeny class is simple but not 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 20 400 3980 78400 1050500 15840400 268421180 4292870400 68719312820 1103550250000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 23 65 303 1025 3863 16385 65503 262145 1052423

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{5}, \sqrt{-11})$$.
Endomorphism algebra over $\overline{\F}_{2^{2}}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.d 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-55})$$$)$
All geometric endomorphisms are defined over $\F_{2^{4}}$.

## 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.a_ad $4$ 2.256.bu_bob