Abelian variety isogeny class 5.2.ah_bb_acv_ft_ajd over $\F_{2}$

• Label: 5.2.ah_bb_acv_ft_ajd
• Base field: $\F_{2}$
• Dimension: $5$
• $p$-rank: $5$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$5$
L-polynomial:	$( 1 - x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )^{2}$
	$1 - 7 x + 27 x^{2} - 73 x^{3} + 149 x^{4} - 237 x^{5} + 298 x^{6} - 292 x^{7} + 216 x^{8} - 112 x^{9} + 32 x^{10}$
Frobenius angles:	$\pm0.123548644961$, $\pm0.123548644961$, $\pm0.384973271919$, $\pm0.456881978294$, $\pm0.456881978294$
Angle rank:	$2$ (numerical)
Jacobians:	$0$

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$5$
Slopes:	$[0, 0, 0, 0, 0, 1, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2$	$2888$	$80864$	$467856$	$20317462$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$10$	$14$	$2$	$16$	$100$	$220$	$322$	$518$	$1090$

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 $\F_{2^{6}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.ab $\times$ 2.2.ad_f^2 and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.2.ab : \(\Q(\sqrt{-7}) \).
• 2.2.ad_f^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{5})\)$)$

Endomorphism algebra over $\overline{\F}_{2}$

The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj $\times$ 1.64.l^4. The endomorphism algebra for each factor is:

• 1.64.aj : \(\Q(\sqrt{-7}) \).
• 1.64.l^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-15}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{2}}$
  The base change of $A$ to $\F_{2^{2}}$ is 1.4.d $\times$ 2.4.b_ad^2. The endomorphism algebra for each factor is:

  • 1.4.d : \(\Q(\sqrt{-7}) \).
  • 2.4.b_ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{5})\)$)$
• Endomorphism algebra over $\F_{2^{3}}$
  The base change of $A$ to $\F_{2^{3}}$ is 1.8.f $\times$ 2.8.a_l^2. The endomorphism algebra for each factor is:

  • 1.8.f : \(\Q(\sqrt{-7}) \).
  • 2.8.a_l^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{5})\)$)$

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
5.2.af_p_abj_cn_adv	$2$	(not in LMFDB)
5.2.ab_d_ab_ab_d	$2$	(not in LMFDB)
5.2.b_d_b_ab_ad	$2$	(not in LMFDB)
5.2.f_p_bj_cn_dv	$2$	(not in LMFDB)
5.2.h_bb_cv_ft_jd	$2$	(not in LMFDB)
5.2.ae_j_an_o_ap	$3$	(not in LMFDB)
5.2.ab_a_c_f_aj	$3$	(not in LMFDB)
5.2.ab_d_ab_ab_d	$3$	(not in LMFDB)
5.2.c_d_f_i_j	$3$	(not in LMFDB)
5.2.f_p_bj_cn_dv	$3$	(not in LMFDB)