Abelian variety isogeny class 1.169.ax over $\F_{13^{2}}$

• Label: 1.169.ax
• Base field: $\F_{13^{2}}$
• Dimension: $1$
• $p$-rank: $1$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{13^{2}}$
Dimension:	$1$
L-polynomial:	$1 - 23 x + 169 x^{2}$
Frobenius angles:	$\pm0.154420958311$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}) \)
Galois group:	$C_2$
Jacobians:	$3$

This isogeny class is simple and 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:	$1$
Slopes:	$[0, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$147$	$28371$	$4826304$	$815751363$	$137859052107$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$147$	$28371$	$4826304$	$815751363$	$137859052107$	$23298094520064$	$3937376507160243$	$665416610388590403$	$112455406959154934976$	$19004963774842628516451$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13^{2}}$.

Endomorphism algebra over $\F_{13^{2}}$

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

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{13^{2}}$.

Subfield	Primitive Model
$\F_{13}$	1.13.ah
$\F_{13}$	1.13.h

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
1.169.x	$2$	(not in LMFDB)
1.169.b	$3$	(not in LMFDB)
1.169.w	$3$	(not in LMFDB)