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

• Label: 2.53.ax_iw
• 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:	$1 - 23 x + 230 x^{2} - 1219 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.0511985002933$, $\pm0.298119617729$
Angle rank:	$2$ (numerical)
Number field:	4.0.814572.2
Galois group:	$D_{4}$
Jacobians:	$4$
Isomorphism classes:	4
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1798$	$7699036$	$22171008544$	$62260687509376$	$174876111723431998$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$31$	$2741$	$148924$	$7890609$	$418168331$	$22163957066$	$1174708119455$	$62259680408833$	$3299763640178908$	$174887471231929421$

Jacobians and polarizations

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

• $y^2=9 x^6+19 x^5+31 x^4+4 x^3+12 x^2+30 x+36$
• $y^2=22 x^6+32 x^5+31 x^4+17 x^3+47 x^2+25 x+45$
• $y^2=51 x^6+31 x^5+49 x^4+38 x^3+34 x^2+47 x+18$
• $y^2=11 x^6+42 x^5+31 x^4+43 x^3+21 x^2+26 x+50$

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 4.0.814572.2.

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_iw	$2$	(not in LMFDB)