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

• Label: 4.4.am_cr_ajm_wu
• 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:	$( 1 - 2 x )^{2}( 1 - 2 x + 4 x^{2} )( 1 - 3 x + 4 x^{2} )^{2}$
	$1 - 12 x + 69 x^{2} - 246 x^{3} + 592 x^{4} - 984 x^{5} + 1104 x^{6} - 768 x^{7} + 256 x^{8}$
Frobenius angles:	$0$, $0$, $\pm0.230053456163$, $\pm0.230053456163$, $\pm0.333333333333$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$12$	$48384$	$21734244$	$5094835200$	$1117190303652$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-7$	$11$	$83$	$303$	$1043$	$3935$	$15827$	$64383$	$260147$	$1045151$

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

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

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey^2 $\times$ 1.4096.bv^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.bv^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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

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

  • 1.64.aq : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.64.j^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
  • 1.64.q : 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.ai_bd_aco_ey	$2$	(not in LMFDB)
4.4.ag_p_as_q	$2$	(not in LMFDB)
4.4.ae_f_s_acm	$2$	(not in LMFDB)
4.4.ac_ab_ag_bg	$2$	(not in LMFDB)
4.4.a_ad_ag_q	$2$	(not in LMFDB)
4.4.a_ad_g_q	$2$	(not in LMFDB)
4.4.c_ab_g_bg	$2$	(not in LMFDB)
4.4.e_f_as_acm	$2$	(not in LMFDB)
4.4.g_p_s_q	$2$	(not in LMFDB)
4.4.i_bd_co_ey	$2$	(not in LMFDB)
4.4.m_cr_jm_wu	$2$	(not in LMFDB)
4.4.ag_j_y_aea	$3$	(not in LMFDB)
4.4.ag_v_abw_dw	$3$	(not in LMFDB)
4.4.ad_d_g_abg	$3$	(not in LMFDB)
4.4.d_ad_am_ai	$3$	(not in LMFDB)
4.4.d_j_y_ca	$3$	(not in LMFDB)
4.4.j_bn_ek_jw	$3$	(not in LMFDB)