Abelian variety isogeny class 2.53.az_jz over $\F_{53}$

• Label: 2.53.az_jz
• Base field: $\F_{53}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{53}$
Dimension:	$2$
L-polynomial:	$1 - 25 x + 259 x^{2} - 1325 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.0599466142417$, $\pm0.237328379738$
Angle rank:	$2$ (numerical)
Number field:	4.0.122525.3
Galois group:	$D_{4}$
Jacobians:	$4$

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1719$	$7596261$	$22138294491$	$62271116769861$	$174891652336898064$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$29$	$2703$	$148703$	$7891931$	$418205494$	$22164304527$	$1174709546263$	$62259674411923$	$3299763501463109$	$174887470298403078$

Jacobians and polarizations

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

• $y^2=32 x^6+29 x^5+28 x^4+20 x^3+26 x^2+17 x+14$
• $y^2=2 x^6+11 x^5+36 x^4+15 x^3+14 x^2+27 x+5$
• $y^2=14 x^6+48 x^5+20 x^4+38 x^3+29 x^2+44 x+35$
• $y^2=10 x^6+22 x^5+40 x^4+44 x^3+14 x^2+3 x+52$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{53}$.

Endomorphism algebra over $\F_{53}$

The endomorphism algebra of this simple isogeny class is 4.0.122525.3.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.53.z_jz	$2$	(not in LMFDB)