Abelian variety isogeny class 2.4.ad_h over $\F_{2^{2}}$

• Label: 2.4.ad_h
• Base field: $\F_{2^{2}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$2$
L-polynomial:	$1 - 3 x + 7 x^{2} - 12 x^{3} + 16 x^{4}$
Frobenius angles:	$\pm0.190783854037$, $\pm0.524117187371$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{13})\)
Galois group:	$C_2^2$
Jacobians:	$4$
Isomorphism classes:	4

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9$	$351$	$4212$	$63531$	$1141299$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$22$	$65$	$250$	$1112$	$4327$	$16382$	$65074$	$262145$	$1048102$

Jacobians and polarizations

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

• $y^2+(x^3+x+1)y=(a+1)x^6+(a+1)x^5+(a+1)x^4+(a+1)x^3+(a+1)x^2+(a+1)x+a+1$
• $y^2+(x^3+x+1)y=(a+1)x^6+ax^5+ax^4+ax^3+(a+1)x^2+ax+a+1$
• $y^2+(x^3+x+1)y=(a+1)x^6+(a+1)x^5+(a+1)x^4+ax^3+(a+1)x^2+ax+a$
• $y^2+(x^3+x+1)y=(a+1)x^6+ax^5+ax^4+(a+1)x^3+(a+1)x^2+(a+1)x+a$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{12}}$.

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

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{13})\).

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

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.el^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-39}) \)$)$

Remainder of endomorphism lattice by field

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

  The base change of $A$ to $\F_{2^{4}}$ is the simple isogeny class 2.16.f_j and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{13})\).

• Endomorphism algebra over $\F_{2^{6}}$

  The base change of $A$ to $\F_{2^{6}}$ is the simple isogeny class 2.64.a_el and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{13})\).

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
2.4.d_h	$2$	2.16.f_j
2.4.a_af	$3$	2.64.a_el
2.4.d_h	$3$	2.64.a_el