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

• Label: 4.3.ah_u_abe_bn
• Base field: $\F_{3}$
• Dimension: $4$
• $p$-rank: $2$
• 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:	$( 1 - 3 x + 3 x^{2} )^{2}( 1 - x - x^{2} - 3 x^{3} + 9 x^{4} )$
	$1 - 7 x + 20 x^{2} - 30 x^{3} + 39 x^{4} - 90 x^{5} + 180 x^{6} - 189 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.126866938441$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.718153680921$
Angle rank:	$2$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5$	$3185$	$356720$	$67283125$	$4435836400$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$1$	$15$	$117$	$302$	$847$	$2531$	$6773$	$19905$	$59296$

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^2 $\times$ 2.3.ab_ab 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^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
• 2.3.ab_ab : 4.0.10933.1.

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc^2 $\times$ 2.729.j_jt. The endomorphism algebra for each factor is:

• 1.729.cc^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 2.729.j_jt : 4.0.10933.1.

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^2 $\times$ 2.9.ad_n. The endomorphism algebra for each factor is:

  • 1.9.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 2.9.ad_n : 4.0.10933.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a^2 $\times$ 2.27.an_dl. The endomorphism algebra for each factor is:

  • 1.27.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 2.27.an_dl : 4.0.10933.1.

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_i_g_abh	$2$	(not in LMFDB)
4.3.ab_ae_a_v	$2$	(not in LMFDB)
4.3.b_ae_a_v	$2$	(not in LMFDB)
4.3.f_i_ag_abh	$2$	(not in LMFDB)
4.3.h_u_be_bn	$2$	(not in LMFDB)
4.3.ae_i_ap_be	$3$	(not in LMFDB)
4.3.ab_f_aj_m	$3$	(not in LMFDB)
4.3.c_c_ad_ag	$3$	(not in LMFDB)