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

• Label: 3.5.ai_bk_adu
• 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: no

Invariants

Base field:	$\F_{5}$
Dimension:	$3$
L-polynomial:	$( 1 - 2 x + 5 x^{2} )( 1 - 3 x + 5 x^{2} )^{2}$
	$1 - 8 x + 36 x^{2} - 98 x^{3} + 180 x^{4} - 200 x^{5} + 125 x^{6}$
Frobenius angles:	$\pm0.265942140215$, $\pm0.265942140215$, $\pm0.352416382350$
Angle rank:	$2$ (numerical)
Jacobians:	$0$

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

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$36$	$23328$	$3068928$	$291600000$	$30840252516$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$34$	$184$	$738$	$3158$	$15244$	$77054$	$389378$	$1953688$	$9771154$

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 $\F_{5}$.

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ad^2 $\times$ 1.5.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.5.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$
• 1.5.ac : \(\Q(\sqrt{-1}) \).

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.ae_m_aw	$2$	3.25.i_dk_pq
3.5.ac_g_ac	$2$	3.25.i_dk_pq
3.5.c_g_c	$2$	3.25.i_dk_pq
3.5.e_m_w	$2$	3.25.i_dk_pq
3.5.i_bk_du	$2$	3.25.i_dk_pq
3.5.b_d_w	$3$	(not in LMFDB)