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

• Label: 5.2.ab_c_a_a_i
• Base field: $\F_{2}$
• Dimension: $5$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2}$
Dimension:	$5$
L-polynomial:	$( 1 - 2 x + 2 x^{2} )( 1 + 2 x^{2} )( 1 + x + 4 x^{5} + 8 x^{6} )$
	$1 - x + 2 x^{2} + 8 x^{5} + 16 x^{8} - 16 x^{9} + 32 x^{10}$
Frobenius angles:	$\pm0.193166924511$, $\pm0.250000000000$, $\pm0.5$, $\pm0.572894129607$, $\pm0.896891574744$
Angle rank:	$3$ (numerical)
Jacobians:	$4$
Isomorphism classes:	672

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$42$	$2520$	$80262$	$932400$	$84121422$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$8$	$14$	$16$	$62$	$104$	$86$	$224$	$446$	$1048$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which 1 is hyperelliptic), and hence is principally polarizable:

• $y^2 + x^3y = x^{11} + x^9 + x^8 + x^6 + x^5 + x^4 + x^2 + x + 1$
• $xz + y^2 + yz = x^2 + xu + y^2 + yt + yu + z^2 + t^2 + u^2 = x^2 + xy + y^2 + yz + z^2 + zt + t^2 = 0$
• $xz + y^2 + yz = xy + xt + xu + yt + yu + zt + zu + t^2 = x^2 + xy + xz + xt + yt + z^2 + zt + zu + u^2 = 0$
• $xz + y^2 + yz = xz + xt + xu + y^2 + yt + yu + zt + t^2 = x^2 + xy + xz + xt + xu + y^2 + yz + yt + yu + z^2 + zt + zu + t^2 + u^2 = 0$

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.b_a_a 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.b_a_a : 6.0.839056.1.

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

The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg^2 $\times$ 3.256.bf_bga_vzc. 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.bf_bga_vzc : 6.0.839056.1.

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.ab_a_i. 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.ab_a_i : 6.0.839056.1.
• 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.ab_q_bg. 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.ab_q_bg : 6.0.839056.1.

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_g_ai_i_ai	$2$	(not in LMFDB)
5.2.b_c_a_a_ai	$2$	(not in LMFDB)
5.2.d_g_i_i_i	$2$	(not in LMFDB)