Abelian variety isogeny class 5.3.b_i_f_bf_q over $\F_{3}$

• Label: 5.3.b_i_f_bf_q
• Base field: $\F_{3}$
• Dimension: $5$
• $p$-rank: $5$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes

Invariants

Base field:	$\F_{3}$
Dimension:	$5$
L-polynomial:	$( 1 + x + 3 x^{2} )( 1 + 5 x^{2} + 16 x^{4} + 45 x^{6} + 81 x^{8} )$
	$1 + x + 8 x^{2} + 5 x^{3} + 31 x^{4} + 16 x^{5} + 93 x^{6} + 45 x^{7} + 216 x^{8} + 81 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.240118583995$, $\pm0.426548082672$, $\pm0.573451917328$, $\pm0.593214749339$, $\pm0.759881416005$
Angle rank:	$1$ (numerical)

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$5$
Slopes:	$[0, 0, 0, 0, 0, 1, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$740$	$328560$	$10952000$	$3785011200$	$951198112700$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$25$	$20$	$89$	$275$	$760$	$2105$	$6449$	$19820$	$57625$

Jacobians and polarizations

This isogeny class is principally polarizable, but it is unknown whether it contains a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{12}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.b $\times$ 4.3.a_f_a_q and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.3.b : \(\Q(\sqrt{-11}) \).
• 4.3.a_f_a_q : 8.0.303595776.1.

Endomorphism algebra over $\overline{\F}_{3}$

The base change of $A$ to $\F_{3^{12}}$ is 1.531441.cag^5 and its endomorphism algebra is $\mathrm{M}_{5}($\(\Q(\sqrt{-11}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{3^{2}}$
  The base change of $A$ to $\F_{3^{2}}$ is 1.9.f $\times$ 2.9.f_q^2. The endomorphism algebra for each factor is:

  • 1.9.f : \(\Q(\sqrt{-11}) \).
  • 2.9.f_q^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{-11})\)$)$
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.ai $\times$ 2.27.a_k^2. The endomorphism algebra for each factor is:

  • 1.27.ai : \(\Q(\sqrt{-11}) \).
  • 2.27.a_k^2 : $\mathrm{M}_{2}($\(\Q(i, \sqrt{11})\)$)$
• Endomorphism algebra over $\F_{3^{4}}$
  The base change of $A$ to $\F_{3^{4}}$ is 1.81.ah $\times$ 2.81.h_abg^2. The endomorphism algebra for each factor is:

  • 1.81.ah : \(\Q(\sqrt{-11}) \).
  • 2.81.h_abg^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{-11})\)$)$
• Endomorphism algebra over $\F_{3^{6}}$
  The base change of $A$ to $\F_{3^{6}}$ is 1.729.ak $\times$ 1.729.k^4. The endomorphism algebra for each factor is:

  • 1.729.ak : \(\Q(\sqrt{-11}) \).
  • 1.729.k^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-11}) \)$)$

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
5.3.ab_i_af_bf_aq	$2$	(not in LMFDB)
5.3.b_ah_ak_n_br	$3$	(not in LMFDB)
5.3.ad_c_af_v_abo	$4$	(not in LMFDB)