Abelian variety isogeny class 5.2.a_a_a_a_e over $\F_{2}$

• Label: 5.2.a_a_a_a_e
• Base field: $\F_{2}$
• Dimension: $5$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2}$
Dimension:	$5$
L-polynomial:	$1 + 4 x^{5} + 32 x^{10}$
Frobenius angles:	$\pm0.123005345616$, $\pm0.276994654384$, $\pm0.523005345616$, $\pm0.676994654384$, $\pm0.923005345616$
Angle rank:	$1$ (numerical)
Number field:	10.0.42017500000000.1
Galois group:	$F_{5}\times C_2$
Jacobians:	$3$
Isomorphism classes:	50

This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$0$
Slopes:	$[2/5, 2/5, 2/5, 2/5, 2/5, 3/5, 3/5, 3/5, 3/5, 3/5]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$37$	$1073$	$32449$	$1048321$	$69343957$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$5$	$9$	$17$	$53$	$65$	$129$	$257$	$513$	$1265$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which 1 is hyperelliptic), and hence is principally polarizable:

• $y^2 + y = x^{11} + x^9 + x^6 + 1$
• $x^5 + x^4y + x^4z + x^3yz + x^3z^2 + x^2yz^2 + xy^4 + xyz^3 + y^4z=0$
• $xt + y^2 + yz + z^2 + zt = x^2 + xu + y^2 + yt + yu + z^2 + t^2 + u^2 = x^2 + xz + y^2 + z^2 + zt + tu + u^2 = 0$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{5}}$.

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is 10.0.42017500000000.1.

Endomorphism algebra over $\overline{\F}_{2}$

The base change of $A$ to $\F_{2^{5}}$ is the simple isogeny class 5.32.u_mi_etc_bntw_jjwy and its endomorphism algebra is the division algebra of dimension 25 over \(\Q(\sqrt{-7}) \) with the following ramification data at primes above $2$, and unramified at all archimedean places:

$v$	($ 2 $,\( \pi \))	($ 2 $,\( \pi + 1 \))
$\operatorname{inv}_v$	$2/5$	$3/5$

where $\pi$ is a root of $x^{2} - x + 2$.

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
5.2.a_a_a_a_ae	$2$	(not in LMFDB)
5.2.a_a_a_a_ae	$10$	(not in LMFDB)