Abelian variety isogeny class 4.5.a_ai_a_bi over $\F_{5}$

• Label: 4.5.a_ai_a_bi
• Base field: $\F_{5}$
• Dimension: $4$
• $p$-rank: $4$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes

Invariants

Base field:	$\F_{5}$
Dimension:	$4$
L-polynomial:	$1 - 8 x^{2} + 34 x^{4} - 200 x^{6} + 625 x^{8}$
Frobenius angles:	$\pm0.0418142407386$, $\pm0.276491818062$, $\pm0.723508181938$, $\pm0.958185759261$
Angle rank:	$2$ (numerical)
Number field:	8.0.110166016.2
Galois group:	$D_4\times C_2$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$4$
Slopes:	$[0, 0, 0, 0, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$452$	$204304$	$239567684$	$153927536896$	$95397053203332$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$10$	$126$	$634$	$3126$	$15034$	$78126$	$387418$	$1953126$	$9771690$

Jacobians and polarizations

This isogeny class is principally polarizable and contains no Jacobian of a hyperelliptic curve, but it is unknown whether it contains a Jacobian of a non-hyperelliptic curve.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

The endomorphism algebra of this simple isogeny class is 8.0.110166016.2.

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

The base change of $A$ to $\F_{5^{2}}$ is 2.25.ai_bi^2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.10496.2$)$

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
4.5.ai_bo_afg_nq	$4$	(not in LMFDB)
4.5.a_i_a_bi	$4$	(not in LMFDB)
4.5.i_bo_fg_nq	$4$	(not in LMFDB)