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

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

Invariants

Base field:	$\F_{5}$
Dimension:	$4$
L-polynomial:	$1 - 7 x + 29 x^{2} - 93 x^{3} + 236 x^{4} - 465 x^{5} + 725 x^{6} - 875 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.0796806907993$, $\pm0.230664863526$, $\pm0.479680690799$, $\pm0.569335136474$
Angle rank:	$2$ (numerical)
Number field:	8.0.58140625.2
Galois group:	$C_2^2:C_4$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$176$	$537856$	$222657776$	$138594734080$	$99370274519296$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$35$	$113$	$567$	$3254$	$15995$	$77713$	$389167$	$1953239$	$9768530$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

The endomorphism algebra of this simple isogeny class is 8.0.58140625.2.

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

The base change of $A$ to $\F_{5^{5}}$ is 2.3125.cm_ecs^2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.1525.1$)$

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
4.5.h_bd_dp_jc	$2$	(not in LMFDB)
4.5.ac_ab_ai_cj	$5$	(not in LMFDB)
4.5.i_bd_cu_gf	$5$	(not in LMFDB)