Abelian variety isogeny class 3.16.au_gu_abim over $\F_{2^{4}}$

• Label: 3.16.au_gu_abim
• Base field: $\F_{2^{4}}$
• Dimension: $3$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2^{4}}$
Dimension:	$3$
L-polynomial:	$( 1 - 4 x )^{4}( 1 - 4 x + 16 x^{2} )$
	$1 - 20 x + 176 x^{2} - 896 x^{3} + 2816 x^{4} - 5120 x^{5} + 4096 x^{6}$
Frobenius angles:	$0$, $0$, $0$, $0$, $\pm0.333333333333$
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:	$[1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1053$	$13820625$	$66556260225$	$278189293370625$	$1147304084679683073$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$209$	$3969$	$64769$	$1043457$	$16752641$	$268353537$	$4294770689$	$68718952449$	$1099508482049$

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 $\F_{2^{24}}$.

Endomorphism algebra over $\F_{2^{4}}$

The isogeny class factors as 1.16.ai^2 $\times$ 1.16.ae and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. 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 $\infty$.
• 1.16.ae : \(\Q(\sqrt{-3}) \).

Endomorphism algebra over $\overline{\F}_{2^{4}}$

The base change of $A$ to $\F_{2^{24}}$ is 1.16777216.amdc^3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{8}}$
  The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg^2 $\times$ 1.256.q. 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.q : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{2^{12}}$
  The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey^2 $\times$ 1.4096.ey. 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.ey : 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
3.16.am_bw_aey	$2$	(not in LMFDB)
3.16.ae_aq_ey	$2$	(not in LMFDB)
3.16.e_aq_aey	$2$	(not in LMFDB)
3.16.m_bw_ey	$2$	(not in LMFDB)
3.16.u_gu_bim	$2$	(not in LMFDB)
3.16.ai_aq_jw	$3$	(not in LMFDB)
3.16.ai_bg_aey	$3$	(not in LMFDB)
3.16.e_aq_aey	$3$	(not in LMFDB)
3.16.e_bg_cm	$3$	(not in LMFDB)
3.16.q_ey_yq	$3$	(not in LMFDB)