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

• Label: 4.3.ai_bh_adn_hb
• 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:	$( 1 - 3 x + 3 x^{2} )( 1 - 5 x + 15 x^{2} - 31 x^{3} + 45 x^{4} - 45 x^{5} + 27 x^{6} )$
	$1 - 8 x + 33 x^{2} - 91 x^{3} + 183 x^{4} - 273 x^{5} + 297 x^{6} - 216 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.113296540390$, $\pm0.166666666667$, $\pm0.351823865540$, $\pm0.481790494592$
Angle rank:	$3$ (numerical)
Isomorphism classes:	1

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7$	$8281$	$732844$	$40585181$	$3533486887$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$12$	$35$	$76$	$246$	$813$	$2369$	$6788$	$20069$	$59632$

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 $\times$ 3.3.af_p_abf 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 : \(\Q(\sqrt{-3}) \).
• 3.3.af_p_abf : 6.0.400967.1.

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc $\times$ 3.729.bd_apb_abvut. The endomorphism algebra for each factor is:

• 1.729.cc : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 3.729.bd_apb_abvut : 6.0.400967.1.

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 $\times$ 3.9.f_f_ah. The endomorphism algebra for each factor is:

  • 1.9.ad : \(\Q(\sqrt{-3}) \).
  • 3.9.f_f_ah : 6.0.400967.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 3.27.h_bn_lb. The endomorphism algebra for each factor is:

  • 1.27.a : \(\Q(\sqrt{-3}) \).
  • 3.27.h_bn_lb : 6.0.400967.1.

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.ac_d_ab_ad	$2$	(not in LMFDB)
4.3.c_d_b_ad	$2$	(not in LMFDB)
4.3.i_bh_dn_hb	$2$	(not in LMFDB)
4.3.af_s_abu_dm	$3$	(not in LMFDB)