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

• Label: 4.4.am_cm_aia_sm
• Base field: $\F_{2^{2}}$
• Dimension: $4$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$4$
L-polynomial:	$( 1 - 2 x )^{6}( 1 + 4 x^{2} )$
	$1 - 12 x + 64 x^{2} - 208 x^{3} + 480 x^{4} - 832 x^{5} + 1024 x^{6} - 768 x^{7} + 256 x^{8}$
Frobenius angles:	$0$, $0$, $0$, $0$, $0$, $0$, $\pm0.5$
Angle rank:	$0$ (numerical)

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5$	$18225$	$7647185$	$2562890625$	$909691273025$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-7$	$1$	$17$	$129$	$833$	$3841$	$15617$	$63489$	$259073$	$1044481$

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 $\F_{2^{8}}$.

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

The isogeny class factors as 1.4.ae^3 $\times$ 1.4.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.4.ae^3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.4.a : \(\Q(\sqrt{-1}) \).

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

The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg^4 and its endomorphism algebra is $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

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^3 $\times$ 1.16.i. The endomorphism algebra for each factor is:

  • 1.16.ai^3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 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.ae_a_q_abg	$2$	(not in LMFDB)
4.4.e_a_aq_abg	$2$	(not in LMFDB)
4.4.m_cm_ia_sm	$2$	(not in LMFDB)
4.4.ag_q_abo_ds	$3$	(not in LMFDB)
4.4.a_e_aq_a	$3$	(not in LMFDB)
4.4.g_bc_dc_hk	$3$	(not in LMFDB)