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

• Label: 5.2.ag_s_abh_bs_ace
• Base field: $\F_{2}$
• Dimension: $5$
• $p$-rank: $3$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$5$
L-polynomial:	$( 1 - 2 x + 2 x^{2} )^{2}( 1 - 2 x + 2 x^{2} - x^{3} + 4 x^{4} - 8 x^{5} + 8 x^{6} )$
	$1 - 6 x + 18 x^{2} - 33 x^{3} + 44 x^{4} - 56 x^{5} + 88 x^{6} - 132 x^{7} + 144 x^{8} - 96 x^{9} + 32 x^{10}$
Frobenius angles:	$\pm0.161334789180$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.327009058845$, $\pm0.739882802642$
Angle rank:	$3$ (numerical)
Jacobians:	$0$

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

Newton polygon

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4$	$2600$	$100048$	$6890000$	$53126324$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$5$	$18$	$49$	$47$	$62$	$123$	$193$	$522$	$1065$

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 $\F_{2^{4}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.ac^2 $\times$ 3.2.ac_c_ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.2.ac^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• 3.2.ac_c_ab : 6.0.2464727.1.

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

The base change of $A$ to $\F_{2^{4}}$ is 1.16.i^2 $\times$ 3.16.q_ey_yp. The endomorphism algebra for each factor is:

• 1.16.i^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 3.16.q_ey_yp : 6.0.2464727.1.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{2}}$
  The base change of $A$ to $\F_{2^{2}}$ is 1.4.a^2 $\times$ 3.4.a_i_ab. The endomorphism algebra for each factor is:

  • 1.4.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 3.4.a_i_ab : 6.0.2464727.1.

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
5.2.ac_c_ab_i_aq	$2$	(not in LMFDB)
5.2.ac_c_b_e_ai	$2$	(not in LMFDB)
5.2.c_c_ab_e_i	$2$	(not in LMFDB)
5.2.c_c_b_i_q	$2$	(not in LMFDB)
5.2.g_s_bh_bs_ce	$2$	(not in LMFDB)
5.2.a_a_d_c_ac	$3$	(not in LMFDB)