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

• Label: 3.5.aj_bp_aek
• 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:	$( 1 - 4 x + 5 x^{2} )( 1 - 3 x + 5 x^{2} )( 1 - 2 x + 5 x^{2} )$
	$1 - 9 x + 41 x^{2} - 114 x^{3} + 205 x^{4} - 225 x^{5} + 125 x^{6}$
Frobenius angles:	$\pm0.147583617650$, $\pm0.265942140215$, $\pm0.352416382350$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	6

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$24$	$17280$	$2600064$	$276480000$	$31024344504$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$27$	$162$	$703$	$3177$	$15552$	$78117$	$391583$	$1956042$	$9768627$

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 $\F_{5^{4}}$.

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ae $\times$ 1.5.ad $\times$ 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.ae : \(\Q(\sqrt{-1}) \).
• 1.5.ad : \(\Q(\sqrt{-11}) \).
• 1.5.ac : \(\Q(\sqrt{-1}) \).

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

The base change of $A$ to $\F_{5^{4}}$ is 1.625.o^2 $\times$ 1.625.bx. The endomorphism algebra for each factor is:

• 1.625.o^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• 1.625.bx : \(\Q(\sqrt{-11}) \).

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{5^{2}}$
  The base change of $A$ to $\F_{5^{2}}$ is 1.25.ag $\times$ 1.25.b $\times$ 1.25.g. The endomorphism algebra for each factor is:

  • 1.25.ag : \(\Q(\sqrt{-1}) \).
  • 1.25.b : \(\Q(\sqrt{-11}) \).
  • 1.25.g : \(\Q(\sqrt{-1}) \).

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.af_n_aba	$2$	3.25.b_bn_o
3.5.ad_f_ag	$2$	3.25.b_bn_o
3.5.ab_b_o	$2$	3.25.b_bn_o
3.5.b_b_ao	$2$	3.25.b_bn_o
3.5.d_f_g	$2$	3.25.b_bn_o
3.5.f_n_ba	$2$	3.25.b_bn_o
3.5.j_bp_ek	$2$	3.25.b_bn_o