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

• Label: 4.3.ai_bh_adm_gy
• Base field: $\F_{3}$
• Dimension: $4$
• $p$-rank: $1$
• 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:	$( 1 - 2 x + 3 x^{2} )( 1 + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}$
	$1 - 8 x + 33 x^{2} - 90 x^{3} + 180 x^{4} - 270 x^{5} + 297 x^{6} - 216 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.5$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8$	$9408$	$834176$	$50878464$	$4336544168$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$12$	$38$	$96$	$296$	$846$	$2264$	$6528$	$19874$	$59532$

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.ac $\times$ 1.3.a 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.ac : \(\Q(\sqrt{-2}) \).
• 1.3.a : \(\Q(\sqrt{-3}) \).

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

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

• 1.729.abu : \(\Q(\sqrt{-2}) \).
• 1.729.cc^3 : $\mathrm{M}_{3}(B)$, 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.c $\times$ 1.9.g. The endomorphism algebra for each factor is:

  • 1.9.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 1.9.c : \(\Q(\sqrt{-2}) \).
  • 1.9.g : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a^3 $\times$ 1.27.k. The endomorphism algebra for each factor is:

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

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.ae_j_as_bk	$2$	(not in LMFDB)
4.3.c_d_a_a	$2$	(not in LMFDB)
4.3.i_bh_dm_gy	$2$	(not in LMFDB)
4.3.al_cf_agy_oo	$3$	(not in LMFDB)
4.3.af_j_a_as	$3$	(not in LMFDB)
4.3.af_s_abt_dm	$3$	(not in LMFDB)
4.3.ac_d_a_a	$3$	(not in LMFDB)
4.3.ac_m_as_cc	$3$	(not in LMFDB)
4.3.b_ad_a_s	$3$	(not in LMFDB)
4.3.b_g_j_s	$3$	(not in LMFDB)
4.3.e_j_s_bk	$3$	(not in LMFDB)
4.3.h_v_bk_cc	$3$	(not in LMFDB)