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

• Label: 3.5.af_k_ar
• 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:	$( 1 - 3 x + 5 x^{2} )( 1 - 2 x - x^{2} - 10 x^{3} + 25 x^{4} )$
	$1 - 5 x + 10 x^{2} - 17 x^{3} + 50 x^{4} - 125 x^{5} + 125 x^{6}$
Frobenius angles:	$\pm0.0190830490162$, $\pm0.265942140215$, $\pm0.685749715683$
Angle rank:	$2$ (numerical)
Jacobians:	$3$
Isomorphism classes:	9

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$39$	$12987$	$1557504$	$257467275$	$30279538419$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$1$	$21$	$100$	$661$	$3101$	$15084$	$77561$	$388421$	$1947412$	$9769101$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which 0 are hyperelliptic), and hence is principally polarizable:

• $2x^4+4x^3z+xz^3+y^4=0$
• $x^4+2x^3y+3x^3z+3x^2y^2+x^2yz+4x^2z^2+xz^3+2y^4+y^2z^2=0$
• $3x^4+x^3y+x^3z+x^2y^2+x^2z^2+xy^3+xz^3+2y^4+y^2z^2=0$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ad $\times$ 2.5.ac_ab 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 : \(\Q(\sqrt{-11}) \).
• 2.5.ac_ab : \(\Q(\zeta_{12})\).

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

The base change of $A$ to $\F_{5^{3}}$ is 1.125.aw^2 $\times$ 1.125.s. The endomorphism algebra for each factor is:

• 1.125.aw^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• 1.125.s : \(\Q(\sqrt{-11}) \).

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.ab_ac_x	$2$	3.25.af_be_ald
3.5.b_ac_ax	$2$	3.25.af_be_ald
3.5.f_k_r	$2$	3.25.af_be_ald
3.5.b_h_ac	$3$	(not in LMFDB)