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

• Label: 5.2.af_p_abh_ci_ado
• 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:	$( 1 - 2 x + 2 x^{2} )( 1 + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 7 x^{3} + 10 x^{4} - 12 x^{5} + 8 x^{6} )$
	$1 - 5 x + 15 x^{2} - 33 x^{3} + 60 x^{4} - 92 x^{5} + 120 x^{6} - 132 x^{7} + 120 x^{8} - 80 x^{9} + 32 x^{10}$
Frobenius angles:	$\pm0.105278500939$, $\pm0.250000000000$, $\pm0.316838792568$, $\pm0.5$, $\pm0.641249159631$
Angle rank:	$3$ (numerical)
Jacobians:	$0$
Isomorphism classes:	21

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6$	$4140$	$35334$	$1407600$	$57965226$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$10$	$10$	$22$	$48$	$58$	$110$	$254$	$532$	$1190$

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 $\F_{2^{8}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.ac $\times$ 1.2.a $\times$ 3.2.ad_f_ah 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 : \(\Q(\sqrt{-1}) \).
• 1.2.a : \(\Q(\sqrt{-2}) \).
• 3.2.ad_f_ah : 6.0.679024.1.

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

The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg^2 $\times$ 3.256.cj_crb_bylz. The endomorphism algebra for each factor is:

• 1.256.abg^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 3.256.cj_crb_bylz : 6.0.679024.1.

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 $\times$ 1.4.e $\times$ 3.4.b_d_af. The endomorphism algebra for each factor is:

  • 1.4.a : \(\Q(\sqrt{-1}) \).
  • 1.4.e : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 3.4.b_d_af : 6.0.679024.1.
• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 1.16.i $\times$ 3.16.f_br_fn. The endomorphism algebra for each factor is:

  • 1.16.ai : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 3.16.f_br_fn : 6.0.679024.1.

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.ab_d_af_i_am	$2$	(not in LMFDB)
5.2.b_d_f_i_m	$2$	(not in LMFDB)
5.2.f_p_bh_ci_do	$2$	(not in LMFDB)