Abelian variety isogeny class 3.5.ag_bb_acq over $\F_{5}$

• Label: 3.5.ag_bb_acq
• Base field: $\F_{5}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5}$
Dimension:	$3$
L-polynomial:	$( 1 - 2 x + 5 x^{2} )^{3}$
	$1 - 6 x + 27 x^{2} - 68 x^{3} + 135 x^{4} - 150 x^{5} + 125 x^{6}$
Frobenius angles:	$\pm0.352416382350$, $\pm0.352416382350$, $\pm0.352416382350$
Angle rank:	$1$ (numerical)
Jacobians:	$3$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$64$	$32768$	$3241792$	$262144000$	$28205509184$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$44$	$192$	$668$	$2880$	$14924$	$77952$	$393788$	$1960320$	$9764204$

Jacobians and polarizations

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

• $y^2=2 x^8+3 x^4+2$
• $x^4+y^4+z^4=0$
• $2 x^4+x^2 y^2+x^2 z^2+y^4+z^4=0$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ac^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$

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
3.5.ac_l_am	$2$	3.25.s_hb_bqy
3.5.c_l_m	$2$	3.25.s_hb_bqy
3.5.g_bb_cq	$2$	3.25.s_hb_bqy
3.5.a_a_w	$3$	(not in LMFDB)