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

• Label: 5.3.al_ch_aia_us_abow
• 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:	$( 1 - 3 x + 3 x^{2} )( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )^{2}$
	$1 - 11 x + 59 x^{2} - 208 x^{3} + 538 x^{4} - 1062 x^{5} + 1614 x^{6} - 1872 x^{7} + 1593 x^{8} - 891 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.0540867239847$, $\pm0.0540867239847$, $\pm0.166666666667$, $\pm0.445913276015$, $\pm0.445913276015$
Angle rank:	$1$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4$	$32368$	$10972528$	$1945705216$	$658181907004$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-7$	$7$	$20$	$35$	$183$	$784$	$2373$	$6507$	$19100$	$58807$

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 $\F_{3^{12}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.ad $\times$ 2.3.ae_i^2 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 : \(\Q(\sqrt{-3}) \).
• 2.3.ae_i^2 : $\mathrm{M}_{2}($\(\Q(\zeta_{8})\)$)$

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

The base change of $A$ to $\F_{3^{12}}$ is 1.531441.acec $\times$ 1.531441.zi^4. The endomorphism algebra for each factor is:

• 1.531441.acec : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 1.531441.zi^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-2}) \)$)$

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 $\times$ 2.9.a_ao^2. The endomorphism algebra for each factor is:

  • 1.9.ad : \(\Q(\sqrt{-3}) \).
  • 2.9.a_ao^2 : $\mathrm{M}_{2}($\(\Q(\zeta_{8})\)$)$
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 2.27.ae_i^2. The endomorphism algebra for each factor is:

  • 1.27.a : \(\Q(\sqrt{-3}) \).
  • 2.27.ae_i^2 : $\mathrm{M}_{2}($\(\Q(\zeta_{8})\)$)$
• Endomorphism algebra over $\F_{3^{4}}$
  The base change of $A$ to $\F_{3^{4}}$ is 1.81.ao^4 $\times$ 1.81.j. The endomorphism algebra for each factor is:

  • 1.81.ao^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-2}) \)$)$
  • 1.81.j : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{3^{6}}$
  The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc $\times$ 2.729.a_zi^2. The endomorphism algebra for each factor is:

  • 1.729.cc : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
  • 2.729.a_zi^2 : $\mathrm{M}_{2}($\(\Q(\zeta_{8})\)$)$

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.af_l_aq_k_g	$2$	(not in LMFDB)
5.3.ad_d_a_ao_bq	$2$	(not in LMFDB)
5.3.d_d_a_ao_abq	$2$	(not in LMFDB)
5.3.f_l_q_k_ag	$2$	(not in LMFDB)
5.3.l_ch_ia_us_bow	$2$	(not in LMFDB)
5.3.ai_bj_aei_ko_aui	$3$	(not in LMFDB)
5.3.af_l_aq_k_g	$3$	(not in LMFDB)
5.3.b_ab_ae_h_bb	$3$	(not in LMFDB)
5.3.e_l_u_bf_bw	$3$	(not in LMFDB)
5.3.h_x_bs_cd_cr	$3$	(not in LMFDB)