# Properties

 Label 2.4.a_b 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: $1 + x^{2} + 16 x^{4}$ Frobenius angles: $\pm0.269946543837$, $\pm0.730053456163$ Angle rank: $1$ (numerical) Number field: $$\Q(i, \sqrt{7})$$ 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 18 324 4050 82944 1049778 16402500 268410258 4236447744 68719950450 1102033849284

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 19 65 319 1025 4003 16385 64639 262145 1050979

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(i, \sqrt{7})$$.
Endomorphism algebra over $\overline{\F}_{2^{2}}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.b 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 $4$ 2.256.ck_cer 2.4.a_ab $4$ 2.256.ck_cer 2.4.g_r $4$ 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 $4$ 2.256.ck_cer 2.4.a_ab $4$ 2.256.ck_cer 2.4.g_r $4$ 2.256.ck_cer 2.4.ad_f $12$ (not in LMFDB) 2.4.d_f $12$ (not in LMFDB)