# Properties

 Label 2.16.aj_br Base Field $\F_{2^{4}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{4}}$ Dimension: $2$ L-polynomial: $1 - 9 x + 43 x^{2} - 144 x^{3} + 256 x^{4}$ Frobenius angles: $\pm0.108303609292$, $\pm0.441636942625$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{37})$$ Galois group: $C_2^2$ Jacobians: 12

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 12 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 147 66591 16780932 4263222411 1098088275837 281599678788624 72072407321352543 18447199763816467539 4722366482733934692252 1208927448569688921723951

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 8 262 4097 65050 1047218 16784647 268490636 4295073394 68719476737 1099513109302

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{37})$$.
Endomorphism algebra over $\overline{\F}_{2^{4}}$
 The base change of $A$ to $\F_{2^{24}}$ is 1.16777216.fmx 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-111})$$$)$
All geometric endomorphisms are defined over $\F_{2^{24}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{8}}$  The base change of $A$ to $\F_{2^{8}}$ is the simple isogeny class 2.256.f_aix and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{37})$$.
• Endomorphism algebra over $\F_{2^{12}}$  The base change of $A$ to $\F_{2^{12}}$ is the simple isogeny class 2.4096.a_fmx and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{37})$$.

## 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 2.16.j_br $2$ 2.256.f_aix 2.16.a_af $3$ (not in LMFDB) 2.16.j_br $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.16.j_br $2$ 2.256.f_aix 2.16.a_af $3$ (not in LMFDB) 2.16.j_br $3$ (not in LMFDB) 2.16.a_af $6$ (not in LMFDB) 2.16.a_f $12$ (not in LMFDB)