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

• Label: 5.2.ah_y_acc_dp_afi
• Base field: $\F_{2}$
• Dimension: $5$
• $p$-rank: $4$
• 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:	$( 1 - 2 x + 2 x^{2} )( 1 - 5 x + 12 x^{2} - 20 x^{3} + 29 x^{4} - 40 x^{5} + 48 x^{6} - 40 x^{7} + 16 x^{8} )$
	$1 - 7 x + 24 x^{2} - 54 x^{3} + 93 x^{4} - 138 x^{5} + 186 x^{6} - 216 x^{7} + 192 x^{8} - 112 x^{9} + 32 x^{10}$
Frobenius angles:	$\pm0.0635622003031$, $\pm0.165221137389$, $\pm0.250000000000$, $\pm0.365221137389$, $\pm0.663562200303$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	1

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

Newton polygon

$p$-rank:	$4$
Slopes:	$[0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$1055$	$24193$	$2220775$	$42917611$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$4$	$8$	$28$	$41$	$46$	$143$	$292$	$476$	$1019$

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 $\F_{2^{20}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.ac $\times$ 4.2.af_m_au_bd 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 : \(\Q(\sqrt{-1}) \).
• 4.2.af_m_au_bd : 8.0.13140625.1.

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

The base change of $A$ to $\F_{2^{20}}$ is 1.1048576.dau $\times$ 2.1048576.dth_ibxft^2. The endomorphism algebra for each factor is:

• 1.1048576.dau : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 2.1048576.dth_ibxft^2 : $\mathrm{M}_{2}($4.0.3625.1$)$

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 $\times$ 4.4.ab_c_ai_z. The endomorphism algebra for each factor is:

  • 1.4.a : \(\Q(\sqrt{-1}) \).
  • 4.4.ab_c_ai_z : 8.0.13140625.1.
• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i $\times$ 4.16.d_bm_bk_yb. The endomorphism algebra for each factor is:

  • 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 4.16.d_bm_bk_yb : 8.0.13140625.1.
• Endomorphism algebra over $\F_{2^{5}}$
  The base change of $A$ to $\F_{2^{5}}$ is 1.32.i $\times$ 4.32.a_ad_a_bwv. The endomorphism algebra for each factor is:

  • 1.32.i : \(\Q(\sqrt{-1}) \).
  • 4.32.a_ad_a_bwv : 8.0.13140625.1.
• Endomorphism algebra over $\F_{2^{10}}$
  The base change of $A$ to $\F_{2^{10}}$ is 1.1024.a $\times$ 2.1024.ad_bwv^2. The endomorphism algebra for each factor is:

  • 1.1024.a : \(\Q(\sqrt{-1}) \).
  • 2.1024.ad_bwv^2 : $\mathrm{M}_{2}($4.0.3625.1$)$

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.ad_e_ag_n_aw	$2$	(not in LMFDB)
5.2.d_e_g_n_w	$2$	(not in LMFDB)
5.2.h_y_cc_dp_fi	$2$	(not in LMFDB)