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

• Label: 2.53.aw_ih
• 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:	$1 - 22 x + 215 x^{2} - 1166 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.0365951132079$, $\pm0.326837590926$
Angle rank:	$2$ (numerical)
Number field:	4.0.62352.1
Galois group:	$D_{4}$
Jacobians:	$6$
Isomorphism classes:	6

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1837$	$7739281$	$22170650128$	$62245187394841$	$174866117878063117$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$32$	$2756$	$148922$	$7888644$	$418144432$	$22163835926$	$1174708296976$	$62259686411908$	$3299763646074770$	$174887470579914596$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=39x^6+17x^5+39x^4+28x^3+52x^2+x+4$
• $y^2=32x^6+34x^5+49x^4+14x^3+20x^2+2x+15$
• $y^2=38x^6+36x^5+34x^4+43x^3+50x^2+8x+50$
• $y^2=32x^6+48x^5+22x^4+5x^3+9x^2+6x+31$
• $y^2=39x^6+12x^5+48x^4+28x^3+24x^2+18x+31$
• $y^2=17x^6+47x^5+52x^4+7x^3+27x^2+52x+38$

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 4.0.62352.1.

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.w_ih	$2$	(not in LMFDB)