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

• Label: 3.3.ae_m_ay
• Base field: $\F_{3}$
• Dimension: $3$
• $p$-rank: $1$
• 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:	$( 1 - 3 x + 3 x^{2} )( 1 - x + 3 x^{2} )( 1 + 3 x^{2} )$
	$1 - 4 x + 12 x^{2} - 24 x^{3} + 36 x^{4} - 36 x^{5} + 27 x^{6}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.406785250661$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$1$
Isomorphism classes:	14

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$12$	$1680$	$28224$	$436800$	$14084412$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$18$	$36$	$66$	$240$	$828$	$2352$	$6594$	$19548$	$58818$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=2x^8+2x^7+2x^6+2x^2+x+2$

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 $\times$ 1.3.ab $\times$ 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 : \(\Q(\sqrt{-3}) \).
• 1.3.ab : \(\Q(\sqrt{-11}) \).
• 1.3.a : \(\Q(\sqrt{-3}) \).

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

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

• 1.729.ak : \(\Q(\sqrt{-11}) \).
• 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 $\times$ 1.9.f $\times$ 1.9.g. The endomorphism algebra for each factor is:

  • 1.9.ad : \(\Q(\sqrt{-3}) \).
  • 1.9.f : \(\Q(\sqrt{-11}) \).
  • 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^2 $\times$ 1.27.i. The endomorphism algebra for each factor is:

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

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
3.3.ac_g_am	$2$	3.9.i_y_cc
3.3.c_g_m	$2$	3.9.i_y_cc
3.3.e_m_y	$2$	3.9.i_y_cc
3.3.ah_y_abz	$3$	(not in LMFDB)
3.3.ab_a_d	$3$	(not in LMFDB)
3.3.ab_j_ag	$3$	(not in LMFDB)
3.3.c_g_m	$3$	(not in LMFDB)
3.3.f_m_v	$3$	(not in LMFDB)