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

• Label: 4.3.ai_bh_ado_he
• Base field: $\F_{3}$
• Dimension: $4$
• $p$-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:	$4$
L-polynomial:	$( 1 - 3 x + 3 x^{2} )( 1 - x + 3 x^{2} )( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )$
	$1 - 8 x + 33 x^{2} - 92 x^{3} + 186 x^{4} - 276 x^{5} + 297 x^{6} - 216 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.0540867239847$, $\pm0.166666666667$, $\pm0.406785250661$, $\pm0.445913276015$
Angle rank:	$2$ (numerical)
Isomorphism classes:	2

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6$	$7140$	$631008$	$31558800$	$2844704886$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$12$	$32$	$56$	$196$	$774$	$2404$	$6688$	$19256$	$58332$

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 $\times$ 1.3.ab $\times$ 2.3.ae_i 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 : \(\Q(\sqrt{-3}) \).
• 1.3.ab : \(\Q(\sqrt{-11}) \).
• 2.3.ae_i : \(\Q(\zeta_{8})\).

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

The base change of $A$ to $\F_{3^{12}}$ is 1.531441.acec $\times$ 1.531441.zi^2 $\times$ 1.531441.cag. The endomorphism algebra for each factor is:

• 1.531441.acec : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 1.531441.zi^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
• 1.531441.cag : \(\Q(\sqrt{-11}) \).

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 $\times$ 1.9.f $\times$ 2.9.a_ao. The endomorphism algebra for each factor is:

  • 1.9.ad : \(\Q(\sqrt{-3}) \).
  • 1.9.f : \(\Q(\sqrt{-11}) \).
  • 2.9.a_ao : \(\Q(\zeta_{8})\).
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 1.27.i $\times$ 2.27.ae_i. The endomorphism algebra for each factor is:

  • 1.27.a : \(\Q(\sqrt{-3}) \).
  • 1.27.i : \(\Q(\sqrt{-11}) \).
  • 2.27.ae_i : \(\Q(\zeta_{8})\).
• Endomorphism algebra over $\F_{3^{4}}$
  The base change of $A$ to $\F_{3^{4}}$ is 1.81.ao^2 $\times$ 1.81.ah $\times$ 1.81.j. The endomorphism algebra for each factor is:

  • 1.81.ao^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
  • 1.81.ah : \(\Q(\sqrt{-11}) \).
  • 1.81.j : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{3^{6}}$
  The base change of $A$ to $\F_{3^{6}}$ is 1.729.ak $\times$ 1.729.cc $\times$ 2.729.a_zi. The endomorphism algebra for each factor is:

  • 1.729.ak : \(\Q(\sqrt{-11}) \).
  • 1.729.cc : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
  • 2.729.a_zi : \(\Q(\zeta_{8})\).

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
4.3.ag_t_abu_dm	$2$	(not in LMFDB)
4.3.ac_d_ac_ag	$2$	(not in LMFDB)
4.3.a_b_ae_ag	$2$	(not in LMFDB)
4.3.a_b_e_ag	$2$	(not in LMFDB)
4.3.c_d_c_ag	$2$	(not in LMFDB)
4.3.g_t_bu_dm	$2$	(not in LMFDB)
4.3.i_bh_do_he	$2$	(not in LMFDB)
4.3.af_s_abv_dm	$3$	(not in LMFDB)
4.3.ac_d_ac_ag	$3$	(not in LMFDB)