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

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

Invariants

Base field:	$\F_{3}$
Dimension:	$5$
L-polynomial:	$( 1 + 3 x + 3 x^{2} )( 1 - 5 x + 11 x^{2} - 14 x^{3} + 17 x^{4} - 42 x^{5} + 99 x^{6} - 135 x^{7} + 81 x^{8} )$
	$1 - 2 x - x^{2} + 4 x^{3} + 8 x^{4} - 33 x^{5} + 24 x^{6} + 36 x^{7} - 27 x^{8} - 162 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.0585233652746$, $\pm0.211357390051$, $\pm0.373008092126$, $\pm0.753920350296$, $\pm0.833333333333$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$91$	$36855$	$13632892$	$5042685375$	$638196694681$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$4$	$26$	$112$	$177$	$760$	$2179$	$6536$	$19781$	$58099$

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

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.d $\times$ 4.3.af_l_ao_r 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.d : \(\Q(\sqrt{-3}) \).
• 4.3.af_l_ao_r : 8.0.285524928949.1.

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

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

• 1.729.cc : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 4.729.ay_ceo_abigz_crind : 8.0.285524928949.1.

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.ad_p_abs_dz. The endomorphism algebra for each factor is:

  • 1.9.ad : \(\Q(\sqrt{-3}) \).
  • 4.9.ad_p_abs_dz : 8.0.285524928949.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 4.27.ac_ak_cd_wd. The endomorphism algebra for each factor is:

  • 1.27.a : \(\Q(\sqrt{-3}) \).
  • 4.27.ac_ak_cd_wd : 8.0.285524928949.1.

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.ai_bd_ack_do_aff	$2$	(not in LMFDB)
5.3.c_ab_ae_i_bh	$2$	(not in LMFDB)
5.3.i_bd_ck_do_ff	$2$	(not in LMFDB)
5.3.ai_bd_ack_do_aff	$3$	(not in LMFDB)
5.3.af_o_abd_by_adg	$3$	(not in LMFDB)