# Properties

 Label 2.4.ad_i Base Field $\F_{2^{2}}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{2}}$ Dimension: $2$ L-polynomial: $( 1 - 3 x + 4 x^{2} )( 1 + 4 x^{2} )$ Frobenius angles: $\pm0.230053456163$, $\pm0.5$ Angle rank: $1$ (numerical) Jacobians: 1

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1]$

## Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

• $y^2+xy=x^5+x^3+(a+1)x^2+x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 10 400 4810 64800 1109050 17508400 267042730 4232347200 68458118170 1100399410000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 2 24 74 256 1082 4272 16298 64576 261146 1049424

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The isogeny class factors as 1.4.ad $\times$ 1.4.a 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 $\times$ 1.16.i. The endomorphism algebra for each factor is: 1.16.ab : $$\Q(\sqrt{-7})$$. 1.16.i : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
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.d_i $2$ 2.16.h_y 2.4.ah_u $4$ 2.256.ab_asm 2.4.ab_ae $4$ 2.256.ab_asm
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.d_i $2$ 2.16.h_y 2.4.ah_u $4$ 2.256.ab_asm 2.4.ab_ae $4$ 2.256.ab_asm 2.4.b_ae $4$ 2.256.ab_asm 2.4.h_u $4$ 2.256.ab_asm 2.4.af_o $12$ (not in LMFDB) 2.4.ab_c $12$ (not in LMFDB) 2.4.b_c $12$ (not in LMFDB) 2.4.f_o $12$ (not in LMFDB)