# Properties

 Label 2.16.aj_bx 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: $1 - 9 x + 49 x^{2} - 144 x^{3} + 256 x^{4}$ Frobenius angles: $\pm0.211195784157$, $\pm0.390536017683$ Angle rank: $2$ (numerical) Number field: 4.0.2873.1 Galois group: $D_{4}$ Jacobians: 15

This isogeny class is simple and 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 15 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 153 70227 17454852 4318188003 1099500203673 281501799061248 72064086198310233 18446948861723384643 4722330513733311484548 1208921450268219685510707

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 8 274 4259 65890 1048568 16778815 268459640 4295014978 68718953315 1099507653874

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.2873.1.
All geometric endomorphisms are defined over $\F_{2^{4}}$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.16.j_bx $2$ 2.256.r_mj