Abelian variety isogeny class 4.4.ae_l_abc_ce over $\F_{2^{2}}$

• Label: 4.4.ae_l_abc_ce
• Base field: $\F_{2^{2}}$
• Dimension: $4$
• $p$-rank: $2$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$4$
L-polynomial:	$( 1 - 2 x )^{2}( 1 + 4 x^{2} )( 1 + 3 x^{2} + 16 x^{4} )$
	$1 - 4 x + 11 x^{2} - 28 x^{3} + 56 x^{4} - 112 x^{5} + 176 x^{6} - 256 x^{7} + 256 x^{8}$
Frobenius angles:	$0$, $0$, $\pm0.311178646770$, $\pm0.5$, $\pm0.688821353230$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$100$	$90000$	$12676300$	$3969000000$	$1034768762500$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$1$	$23$	$49$	$239$	$961$	$3863$	$16129$	$64479$	$261121$	$1052423$

Jacobians and polarizations

This isogeny class is principally polarizable, but it is unknown whether it contains a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{8}}$.

Endomorphism algebra over $\F_{2^{2}}$

The isogeny class factors as 1.4.ae $\times$ 1.4.a $\times$ 2.4.a_d and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.4.ae : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.4.a : \(\Q(\sqrt{-1}) \).
• 2.4.a_d : \(\Q(\sqrt{5}, \sqrt{-11})\).

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

The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg^2 $\times$ 1.256.x^2. 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 $\infty$.
• 1.256.x^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-55}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 1.16.d^2 $\times$ 1.16.i. The endomorphism algebra for each factor is:

  • 1.16.ai : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.16.d^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-55}) \)$)$
  • 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

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
4.4.e_l_bc_ce	$2$	(not in LMFDB)
4.4.c_l_o_ce	$3$	(not in LMFDB)
4.4.ai_v_ai_abo	$4$	(not in LMFDB)