# Properties

 Label 2.4.ab_g 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: $( 1 - 2 x + 4 x^{2} )( 1 + x + 4 x^{2} )$ Frobenius angles: $\pm0.333333333333$, $\pm0.580430623255$ Angle rank: $1$ (numerical) Jacobians: 2

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 2 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 18 504 4374 65520 1078398 16288776 262290438 4326547680 69193972494 1098831485304

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 28 70 256 1054 3976 16006 66016 263950 1047928

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The isogeny class factors as 1.4.ac $\times$ 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:
Endomorphism algebra over $\overline{\F}_{2^{2}}$
 The base change of $A$ to $\F_{2^{6}}$ is 1.64.al $\times$ 1.64.q. The endomorphism algebra for each factor is: 1.64.al : $$\Q(\sqrt{-15})$$. 1.64.q : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
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.ad_k $2$ 2.16.l_ci 2.4.b_g $2$ 2.16.l_ci 2.4.d_k $2$ 2.16.l_ci 2.4.f_m $3$ 2.64.f_abw
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.ad_k $2$ 2.16.l_ci 2.4.b_g $2$ 2.16.l_ci 2.4.d_k $2$ 2.16.l_ci 2.4.f_m $3$ 2.64.f_abw 2.4.af_m $6$ (not in LMFDB) 2.4.ad_e $6$ (not in LMFDB) 2.4.d_e $6$ (not in LMFDB) 2.4.ab_i $12$ (not in LMFDB) 2.4.b_i $12$ (not in LMFDB)