Abelian variety isogeny class 4.3.ai_bi_ads_hn over $\F_{3}$

• Label: 4.3.ai_bi_ads_hn
• Base field: $\F_{3}$
• Dimension: $4$
• $p$-rank: $2$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3}$
Dimension:	$4$
L-polynomial:	$( 1 - 3 x + 3 x^{2} )^{2}( 1 - x + 3 x^{2} )^{2}$
	$1 - 8 x + 34 x^{2} - 96 x^{3} + 195 x^{4} - 288 x^{5} + 306 x^{6} - 216 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.166666666667$, $\pm0.406785250661$, $\pm0.406785250661$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9$	$11025$	$1016064$	$46580625$	$3331944729$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$14$	$44$	$86$	$236$	$818$	$2516$	$6950$	$19412$	$57614$

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 $\times$ 1.3.ab^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^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
• 1.3.ab^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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

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

• 1.729.ak^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$
• 1.729.cc^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.

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 $\times$ 1.9.f^2. The endomorphism algebra for each factor is:

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

  • 1.27.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 1.27.i^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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
4.3.ag_u_abw_dp	$2$	(not in LMFDB)
4.3.ae_k_ay_bz	$2$	(not in LMFDB)
4.3.ac_e_a_ad	$2$	(not in LMFDB)
4.3.a_c_a_d	$2$	(not in LMFDB)
4.3.c_e_a_ad	$2$	(not in LMFDB)
4.3.e_k_y_bz	$2$	(not in LMFDB)
4.3.g_u_bw_dp	$2$	(not in LMFDB)
4.3.i_bi_ds_hn	$2$	(not in LMFDB)
4.3.af_h_m_abw	$3$	(not in LMFDB)
4.3.af_t_abw_ds	$3$	(not in LMFDB)
4.3.ac_b_g_am	$3$	(not in LMFDB)
4.3.ac_n_as_ci	$3$	(not in LMFDB)
4.3.b_af_a_y	$3$	(not in LMFDB)
4.3.b_e_j_g	$3$	(not in LMFDB)
4.3.b_h_m_y	$3$	(not in LMFDB)
4.3.e_h_m_y	$3$	(not in LMFDB)
4.3.h_t_y_y	$3$	(not in LMFDB)