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

• Label: 2.53.ax_jd
• 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:	$1 - 23 x + 237 x^{2} - 1219 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.166294039069$, $\pm0.247319436379$
Angle rank:	$2$ (numerical)
Number field:	4.0.137525.1
Galois group:	$D_{4}$
Jacobians:	$5$
Isomorphism classes:	5

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1805$	$7741645$	$22243413905$	$62326009775525$	$174914827268512000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$31$	$2755$	$149407$	$7898883$	$418260906$	$22164673915$	$1174711586107$	$62259682684323$	$3299763505857091$	$174887469885474150$

Jacobians and polarizations

This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=x^6+2x^5+51x^4+41x^3+28x^2+5x+3$
• $y^2=3x^6+38x^5+14x^4+15x^3+39x^2+14x+51$
• $y^2=12x^6+35x^5+32x^4+29x^3+10x^2+9$
• $y^2=14x^6+17x^5+30x^3+20x^2+10x+35$
• $y^2=50x^6+29x^5+41x^4+43x^3+20x^2+14x+48$

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 4.0.137525.1.

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.x_jd	$2$	(not in LMFDB)