Abelian variety isogeny class 3.3.ae_e_a over $\F_{3}$

• Label: 3.3.ae_e_a
• Base field: $\F_{3}$
• Dimension: $3$
• $p$-rank: $2$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3}$
Dimension:	$3$
L-polynomial:	$( 1 - 3 x + 3 x^{2} )( 1 - x - 2 x^{2} - 3 x^{3} + 9 x^{4} )$
	$1 - 4 x + 4 x^{2} + 12 x^{4} - 36 x^{5} + 27 x^{6}$
Frobenius angles:	$\pm0.0734519173280$, $\pm0.166666666667$, $\pm0.740118583995$
Angle rank:	$1$ (numerical)
Jacobians:	$1$
Isomorphism classes:	4

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4$	$336$	$11200$	$646464$	$14147284$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$2$	$12$	$98$	$240$	$764$	$2352$	$6530$	$19956$	$59282$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is not hyperelliptic), and hence is principally polarizable:

• $x^4+2x^3z+x^2y^2+x^2z^2+xy^2z+2y^4+y^3z+z^4=0$

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 $\times$ 2.3.ab_ac 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.ab_ac : \(\Q(\sqrt{-3}, \sqrt{-11})\).

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

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

• 1.729.ak^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$
• 1.729.cc : 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 $\times$ 2.9.af_q. The endomorphism algebra for each factor is:

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

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

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
3.3.ac_ac_m	$2$	3.9.ai_bo_afi
3.3.c_ac_am	$2$	3.9.ai_bo_afi
3.3.e_e_a	$2$	3.9.ai_bo_afi
3.3.ab_b_ag	$3$	(not in LMFDB)
3.3.ab_e_aj	$3$	(not in LMFDB)
3.3.c_ac_am	$3$	(not in LMFDB)
3.3.c_k_m	$3$	(not in LMFDB)
3.3.f_q_bh	$3$	(not in LMFDB)