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

• Label: 4.4.al_ci_aia_tc
• Base field: $\F_{2^{2}}$
• Dimension: $4$
• $p$-rank: $1$
• 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:	$( 1 - 2 x )^{2}( 1 - 3 x + 4 x^{2} )( 1 - 2 x + 4 x^{2} )^{2}$
	$1 - 11 x + 60 x^{2} - 208 x^{3} + 496 x^{4} - 832 x^{5} + 960 x^{6} - 704 x^{7} + 256 x^{8}$
Frobenius angles:	$0$, $0$, $\pm0.230053456163$, $\pm0.333333333333$, $\pm0.333333333333$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$18$	$63504$	$23790186$	$4829479200$	$1025295722298$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$16$	$90$	$288$	$954$	$3760$	$15786$	$65088$	$262170$	$1047376$

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 $\times$ 1.4.ad $\times$ 1.4.ac^2 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.ad : \(\Q(\sqrt{-7}) \).
• 1.4.ac^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey^3 $\times$ 1.4096.bv. The endomorphism algebra for each factor is:

• 1.4096.aey^3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.4096.bv : \(\Q(\sqrt{-7}) \).

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.ab $\times$ 1.16.e^2. The endomorphism algebra for each factor is:

  • 1.16.ai : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.16.ab : \(\Q(\sqrt{-7}) \).
  • 1.16.e^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aq $\times$ 1.64.j $\times$ 1.64.q^2. The endomorphism algebra for each factor is:

  • 1.64.aq : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.64.j : \(\Q(\sqrt{-7}) \).
  • 1.64.q^2 : $\mathrm{M}_{2}(B)$, where $B$ is 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.ah_y_ace_ei	$2$	(not in LMFDB)
4.4.af_m_aq_q	$2$	(not in LMFDB)
4.4.ad_e_aq_bw	$2$	(not in LMFDB)
4.4.ad_e_q_abw	$2$	(not in LMFDB)
4.4.ab_a_ai_q	$2$	(not in LMFDB)
4.4.b_a_i_q	$2$	(not in LMFDB)
4.4.d_e_aq_abw	$2$	(not in LMFDB)
4.4.d_e_q_bw	$2$	(not in LMFDB)
4.4.f_m_q_q	$2$	(not in LMFDB)
4.4.h_y_ce_ei	$2$	(not in LMFDB)
4.4.l_ci_ia_tc	$2$	(not in LMFDB)
4.4.af_g_u_adc	$3$	(not in LMFDB)
4.4.af_s_abo_dk	$3$	(not in LMFDB)
4.4.b_am_ae_cm	$3$	(not in LMFDB)
4.4.b_a_i_q	$3$	(not in LMFDB)
4.4.h_s_u_q	$3$	(not in LMFDB)