Abelian variety isogeny class 2.5.ac_ab over $\F_{5}$

• Label: 2.5.ac_ab
• Base field: $\F_{5}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: no
• Contains a Jacobian: no

Invariants

Base field:	$\F_{5}$
Dimension:	$2$
L-polynomial:	$1 - 2 x - x^{2} - 10 x^{3} + 25 x^{4}$
Frobenius angles:	$\pm0.0190830490162$, $\pm0.685749715683$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\zeta_{12})\)
Galois group:	$C_2^2$
Jacobians:	$0$
Isomorphism classes:	2

This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$13$	$481$	$10816$	$381433$	$9512893$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$20$	$82$	$612$	$3044$	$15158$	$78068$	$389572$	$1948330$	$9766100$

Jacobians and polarizations

This isogeny class is not principally polarizable, and therefore does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{3}}$.

Endomorphism algebra over $\F_{5}$

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{12})\).

Endomorphism algebra over $\overline{\F}_{5}$

The base change of $A$ to $\F_{5^{3}}$ is 1.125.aw^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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
2.5.c_ab	$2$	2.25.ag_l
2.5.e_o	$3$	2.125.abs_bcg
2.5.ae_l	$4$	2.625.ao_aqn