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

• Label: 4.3.ah_y_acd_dy
• 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:	$( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )( 1 - 2 x + 2 x^{2} - 6 x^{3} + 9 x^{4} )$
	$1 - 7 x + 24 x^{2} - 55 x^{3} + 102 x^{4} - 165 x^{5} + 216 x^{6} - 189 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.116139763599$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.616139763599$
Angle rank:	$2$ (numerical)
Isomorphism classes:	22

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8$	$6720$	$463904$	$55910400$	$4528043608$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$9$	$24$	$101$	$307$	$738$	$2209$	$6925$	$20112$	$59289$

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.ac $\times$ 2.3.ac_c 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.ac : \(\Q(\sqrt{-2}) \).
• 2.3.ac_c : \(\Q(i, \sqrt{5})\).

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

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

• 1.531441.acec : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 1.531441.azi : \(\Q(\sqrt{-2}) \).
• 1.531441.sk^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$

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.c $\times$ 2.9.a_ac. The endomorphism algebra for each factor is:

  • 1.9.ad : \(\Q(\sqrt{-3}) \).
  • 1.9.c : \(\Q(\sqrt{-2}) \).
  • 2.9.a_ac : \(\Q(i, \sqrt{5})\).
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 1.27.k $\times$ 2.27.ao_du. The endomorphism algebra for each factor is:

  • 1.27.a : \(\Q(\sqrt{-3}) \).
  • 1.27.k : \(\Q(\sqrt{-2}) \).
  • 2.27.ao_du : \(\Q(i, \sqrt{5})\).
• Endomorphism algebra over $\F_{3^{4}}$
  The base change of $A$ to $\F_{3^{4}}$ is 1.81.ac^2 $\times$ 1.81.j $\times$ 1.81.o. The endomorphism algebra for each factor is:

  • 1.81.ac^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$
  • 1.81.j : \(\Q(\sqrt{-3}) \).
  • 1.81.o : \(\Q(\sqrt{-2}) \).
• Endomorphism algebra over $\F_{3^{6}}$
  The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.cc $\times$ 2.729.a_sk. The endomorphism algebra for each factor is:

  • 1.729.abu : \(\Q(\sqrt{-2}) \).
  • 1.729.cc : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
  • 2.729.a_sk : \(\Q(i, \sqrt{5})\).

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.ad_e_al_be	$2$	(not in LMFDB)
4.3.ad_e_f_as	$2$	(not in LMFDB)
4.3.ab_a_ab_g	$2$	(not in LMFDB)
4.3.b_a_b_g	$2$	(not in LMFDB)
4.3.d_e_af_as	$2$	(not in LMFDB)
4.3.d_e_l_be	$2$	(not in LMFDB)
4.3.h_y_cd_dy	$2$	(not in LMFDB)
4.3.ae_m_abc_cc	$3$	(not in LMFDB)