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

• Label: 4.3.aj_bq_aev_jy
• 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:	$( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )^{3}$
	$1 - 9 x + 42 x^{2} - 125 x^{3} + 258 x^{4} - 375 x^{5} + 378 x^{6} - 243 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.304086723985$
Angle rank:	$1$ (numerical)

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8$	$12096$	$1536416$	$80510976$	$3840744248$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$13$	$58$	$133$	$265$	$646$	$2011$	$6541$	$20254$	$60253$

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 $\times$ 1.3.ac^3 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^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-2}) \)$)$

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

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

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

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

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

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.af_o_az_bq	$2$	(not in LMFDB)
4.3.ad_g_b_ag	$2$	(not in LMFDB)
4.3.ab_c_af_s	$2$	(not in LMFDB)
4.3.b_c_f_s	$2$	(not in LMFDB)
4.3.d_g_ab_ag	$2$	(not in LMFDB)
4.3.f_o_z_bq	$2$	(not in LMFDB)
4.3.j_bq_ev_jy	$2$	(not in LMFDB)
4.3.ag_y_ack_ew	$3$	(not in LMFDB)
4.3.ad_d_k_abe	$3$	(not in LMFDB)
4.3.a_d_k_a	$3$	(not in LMFDB)
4.3.d_d_k_be	$3$	(not in LMFDB)