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

• Label: 5.3.ak_by_agk_pu_abep
• Base field: $\F_{3}$
• Dimension: $5$
• $p$-rank: $4$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3}$
Dimension:	$5$
L-polynomial:	$( 1 - 3 x + 3 x^{2} )( 1 - 7 x + 26 x^{2} - 67 x^{3} + 131 x^{4} - 201 x^{5} + 234 x^{6} - 189 x^{7} + 81 x^{8} )$
	$1 - 10 x + 50 x^{2} - 166 x^{3} + 410 x^{4} - 795 x^{5} + 1230 x^{6} - 1494 x^{7} + 1350 x^{8} - 810 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.0365491909691$, $\pm0.166666666667$, $\pm0.234353293111$, $\pm0.355928212636$, $\pm0.548250857189$
Angle rank:	$4$ (numerical)

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

Newton polygon

$p$-rank:	$4$
Slopes:	$[0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9$	$59031$	$15595524$	$3463289739$	$933273474489$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$10$	$30$	$82$	$269$	$742$	$2080$	$6482$	$19776$	$58315$

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 $\F_{3^{6}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.ad $\times$ 4.3.ah_ba_acp_fb 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.ad : \(\Q(\sqrt{-3}) \).
• 4.3.ah_ba_acp_fb : 8.0.371495353.1.

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc $\times$ 4.729.abq_boh_acfrw_djcin. The endomorphism algebra for each factor is:

• 1.729.cc : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 4.729.abq_boh_acfrw_djcin : 8.0.371495353.1.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{3^{2}}$
  The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad $\times$ 4.9.d_a_ax_adl. The endomorphism algebra for each factor is:

  • 1.9.ad : \(\Q(\sqrt{-3}) \).
  • 4.9.d_a_ax_adl : 8.0.371495353.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 4.27.c_at_ak_ml. The endomorphism algebra for each factor is:

  • 1.27.a : \(\Q(\sqrt{-3}) \).
  • 4.27.c_at_ak_ml : 8.0.371495353.1.

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
5.3.ae_i_ak_i_aj	$2$	(not in LMFDB)
5.3.e_i_k_i_j	$2$	(not in LMFDB)
5.3.k_by_gk_pu_bep	$2$	(not in LMFDB)
5.3.ah_bd_adk_ib_apm	$3$	(not in LMFDB)