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

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

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$4$
L-polynomial:	$( 1 - 2 x )^{4}( 1 - 3 x + 7 x^{2} - 12 x^{3} + 16 x^{4} )$
	$1 - 11 x + 55 x^{2} - 172 x^{3} + 392 x^{4} - 688 x^{5} + 880 x^{6} - 704 x^{7} + 256 x^{8}$
Frobenius angles:	$0$, $0$, $0$, $0$, $\pm0.190783854037$, $\pm0.524117187371$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9$	$28431$	$10113012$	$3216256875$	$1054013593779$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$6$	$33$	$186$	$984$	$4071$	$15870$	$64050$	$260097$	$1044006$

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 $\F_{2^{12}}$.

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

The isogeny class factors as 1.4.ae^2 $\times$ 2.4.ad_h 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^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 2.4.ad_h : \(\Q(\sqrt{-3}, \sqrt{13})\).

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

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey^2 $\times$ 1.4096.el^2. The endomorphism algebra for each factor is:

• 1.4096.aey^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.4096.el^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-39}) \)$)$

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^2 $\times$ 2.16.f_j. The endomorphism algebra for each factor is:

  • 1.16.ai^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 2.16.f_j : \(\Q(\sqrt{-3}, \sqrt{13})\).
• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aq^2 $\times$ 2.64.a_el. The endomorphism algebra for each factor is:

  • 1.64.aq^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 2.64.a_el : \(\Q(\sqrt{-3}, \sqrt{13})\).

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.af_h_ae_i	$2$	(not in LMFDB)
4.4.ad_ab_m_ay	$2$	(not in LMFDB)
4.4.d_ab_am_ay	$2$	(not in LMFDB)
4.4.f_h_e_i	$2$	(not in LMFDB)
4.4.l_cd_gq_pc	$2$	(not in LMFDB)
4.4.ai_t_i_adk	$3$	(not in LMFDB)
4.4.af_n_abi_dc	$3$	(not in LMFDB)
4.4.ac_af_c_bg	$3$	(not in LMFDB)
4.4.b_b_ak_aq	$3$	(not in LMFDB)
4.4.b_h_ae_u	$3$	(not in LMFDB)
4.4.e_h_ae_abc	$3$	(not in LMFDB)
4.4.h_bf_do_ie	$3$	(not in LMFDB)