Abelian variety isogeny class 2.4.ad_f over F22\F_{2^{2}}

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

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$2$
L-polynomial:	$1 - 3 x + 5 x^{2} - 12 x^{3} + 16 x^{4}$
Frobenius angles:	$\pm0.103279877171$, $\pm0.563386789496$
Angle rank:	$1$ (numerical)
Number field:	$\Q(\sqrt{-3}, \sqrt{-7})$
Galois group:	$C_2^2$
Jacobians:	$2$

This isogeny class is simple but not geometrically simple, not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7$	$259$	$3136$	$58275$	$1109227$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$18$	$47$	$226$	$1082$	$4191$	$16298$	$65986$	$264143$	$1049778$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

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

where $a$ is a root of the Conway polynomial.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{6}}$.

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

The endomorphism algebra of this simple isogeny class is $\Q(\sqrt{-3}, \sqrt{-7})$.

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

The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj^2 and its endomorphism algebra is $\mathrm{M}_{2}($$\Q(\sqrt{-7})$$)$

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.

Subfield	Primitive Model
$\F_{2}$	2.2.ab_ab
$\F_{2}$	2.2.b_ab

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.4.d_f	$2$	2.16.b_ap
2.4.g_r	$3$	2.64.as_ib
2.4.ag_r	$6$	(not in LMFDB)