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

• Label: 5.2.d_d_ad_aj_ao
• 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:	$( 1 + 2 x + 2 x^{2} )( 1 + x - x^{2} - 3 x^{3} - x^{4} - 6 x^{5} - 4 x^{6} + 8 x^{7} + 16 x^{8} )$
	$1 + 3 x + 3 x^{2} - 3 x^{3} - 9 x^{4} - 14 x^{5} - 18 x^{6} - 12 x^{7} + 24 x^{8} + 48 x^{9} + 32 x^{10}$
Frobenius angles:	$\pm0.0150267280813$, $\pm0.415026728081$, $\pm0.750000000000$, $\pm0.784973271919$, $\pm0.815026728081$
Angle rank:	$1$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$55$	$605$	$11605$	$1636525$	$5856400$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$2$	$0$	$26$	$-19$	$74$	$132$	$194$	$540$	$797$

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.c $\times$ 4.2.b_ab_ad_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.c : \(\Q(\sqrt{-1}) \).
• 4.2.b_ab_ad_ab : 8.0.37515625.1.

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

The base change of $A$ to $\F_{2^{20}}$ is 1.1048576.abuf^4 $\times$ 1.1048576.dau. The endomorphism algebra for each factor is:

• 1.1048576.abuf^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-7}) \)$)$
• 1.1048576.dau : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

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.ad_f_ad_al. The endomorphism algebra for each factor is:

  • 1.4.a : \(\Q(\sqrt{-1}) \).
  • 4.4.ad_f_ad_al : 8.0.37515625.1.
• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i $\times$ 4.16.b_ap_abf_ib. The endomorphism algebra for each factor is:

  • 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 4.16.b_ap_abf_ib : 8.0.37515625.1.
• Endomorphism algebra over $\F_{2^{5}}$
  The base change of $A$ to $\F_{2^{5}}$ is 1.32.al^4 $\times$ 1.32.ai. The endomorphism algebra for each factor is:

  • 1.32.al^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-7}) \)$)$
  • 1.32.ai : \(\Q(\sqrt{-1}) \).
• Endomorphism algebra over $\F_{2^{10}}$
  The base change of $A$ to $\F_{2^{10}}$ is 1.1024.acf^4 $\times$ 1.1024.a. The endomorphism algebra for each factor is:

  • 1.1024.acf^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-7}) \)$)$
  • 1.1024.a : \(\Q(\sqrt{-1}) \).

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_d_d_aj_o	$2$	(not in LMFDB)
5.2.ab_ab_b_d_ak	$2$	(not in LMFDB)
5.2.b_ab_ab_d_k	$2$	(not in LMFDB)