Abelian variety isogeny class 2.16.ai_bg over $\F_{2^{4}}$

• Label: 2.16.ai_bg
• Base field: $\F_{2^{4}}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{4}}$
Dimension:	$2$
L-polynomial:	$( 1 - 4 x )^{2}( 1 + 16 x^{2} )$
	$1 - 8 x + 32 x^{2} - 128 x^{3} + 256 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.5$
Angle rank:	$0$ (numerical)
Jacobians:	$4$

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$153$	$65025$	$16260993$	$4228250625$	$1097366239233$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$257$	$3969$	$64513$	$1046529$	$16777217$	$268402689$	$4294705153$	$68718952449$	$1099511627777$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2+y=(a^3+a^2+a)x^5+ax^4+(a^3+a^2+a)x^3+a^3+a+1$
• $y^2+y=(a^3+a^2+1)x^5+(a^3+1)x^4+(a^3+a^2+1)x^3$
• $y^2+y=(a^3+1)x^5+(a+1)x^4+(a^3+1)x^3+a^3+a^2+1$
• $y^2+y=(a^3+a+1)x^5+a^2x^4+(a^3+a+1)x^3+a^3+1$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{16}}$.

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

The isogeny class factors as 1.16.ai $\times$ 1.16.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.16.ai : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.16.a : \(\Q(\sqrt{-1}) \).

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

The base change of $A$ to $\F_{2^{16}}$ is 1.65536.ats^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, 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 $\times$ 1.256.bg. The endomorphism algebra for each factor is:

  • 1.256.abg : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.256.bg : 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
2.16.i_bg	$2$	2.256.a_ats
2.16.e_bg	$3$	(not in LMFDB)
2.16.aq_ds	$4$	(not in LMFDB)