# Properties

 Label 2.4.ab_ad 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: $1 - x - 3 x^{2} - 4 x^{3} + 16 x^{4}$ Frobenius angles: $\pm0.0862360434115$, $\pm0.752902710078$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Galois group: $C_2^2$ Jacobians: 1

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 1 curves, and hence is principally polarizable:

• $y^2+(x^3+x+1)y=ax^6+x^5+x^4+x^3+ax^2+x+a$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 9 171 2916 69939 988749 16842816 272594709 4280336739 69130081476 1101267647451

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 10 43 274 964 4111 16636 65314 263707 1050250

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{5})$$.
Endomorphism algebra over $\overline{\F}_{2^{2}}$
 The base change of $A$ to $\F_{2^{6}}$ is 1.64.al 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$
All geometric endomorphisms are defined over $\F_{2^{6}}$.

## 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.b_ad $2$ 2.16.ah_bh 2.4.c_j $3$ 2.64.aw_jp 2.4.ac_j $6$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.b_ad $2$ 2.16.ah_bh 2.4.c_j $3$ 2.64.aw_jp 2.4.ac_j $6$ (not in LMFDB) 2.4.a_h $6$ (not in LMFDB) 2.4.a_ah $12$ (not in LMFDB)