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

• Label: 5.3.ak_bx_agb_ol_abbp
• Base field: $\F_{3}$
• Dimension: $5$
• $p$-rank: $3$
• 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:	$( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 10 x^{2} - 19 x^{3} + 30 x^{4} - 36 x^{5} + 27 x^{6} )$
	$1 - 10 x + 49 x^{2} - 157 x^{3} + 375 x^{4} - 717 x^{5} + 1125 x^{6} - 1413 x^{7} + 1323 x^{8} - 810 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.125412673718$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.335294135736$, $\pm0.584823404300$
Angle rank:	$3$ (numerical)

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

Newton polygon

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9$	$56007$	$14881104$	$4514892291$	$1174205479779$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$8$	$27$	$100$	$324$	$821$	$2304$	$7060$	$20412$	$58688$

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^2 $\times$ 3.3.ae_k_at 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^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
• 3.3.ae_k_at : 6.0.11822771.1.

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc^2 $\times$ 3.729.ar_fe_abgaf. The endomorphism algebra for each factor is:

• 1.729.cc^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 3.729.ar_fe_abgaf : 6.0.11822771.1.

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^2 $\times$ 3.9.e_i_f. The endomorphism algebra for each factor is:

  • 1.9.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 3.9.e_i_f : 6.0.11822771.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a^2 $\times$ 3.27.ab_ai_jr. The endomorphism algebra for each factor is:

  • 1.27.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 3.27.ab_ai_jr : 6.0.11822771.1.

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_h_ah_j_ap	$2$	(not in LMFDB)
5.3.ac_b_b_d_ad	$2$	(not in LMFDB)
5.3.c_b_ab_d_d	$2$	(not in LMFDB)
5.3.e_h_h_j_p	$2$	(not in LMFDB)
5.3.k_bx_gb_ol_bbp	$2$	(not in LMFDB)
5.3.ah_bc_ade_hk_aoc	$3$	(not in LMFDB)
5.3.ae_h_ah_j_ap	$3$	(not in LMFDB)
5.3.ae_q_abr_dv_ahe	$3$	(not in LMFDB)
5.3.ab_e_ae_g_ag	$3$	(not in LMFDB)
5.3.c_b_ab_d_d	$3$	(not in LMFDB)