Properties

 Label 2.4.a_ad 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: $1 - 3 x^{2} + 16 x^{4}$ Frobenius angles: $\pm0.188821353230$, $\pm0.811178646770$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-5}, \sqrt{11})$$ Galois group: $C_2^2$ Jacobians: 2

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 2 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 14 196 4214 78400 1046654 17757796 268449734 4292870400 68719640654 1095484595716

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 11 65 303 1025 4331 16385 65503 262145 1044731

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-5}, \sqrt{11})$$.
Endomorphism algebra over $\overline{\F}_{2^{2}}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.ad 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-55})$$$)$
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 2.4.a_d $4$ 2.256.bu_bob