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

• Label: 2.53.aw_in
• 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:	$1 - 22 x + 221 x^{2} - 1166 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.125136993969$, $\pm0.300209973469$
Angle rank:	$2$ (numerical)
Number field:	4.0.2488896.4
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1843$	$7775617$	$22229949004$	$62295576367369$	$174893994213097123$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$32$	$2768$	$149318$	$7895028$	$418211092$	$22164330638$	$1174711086532$	$62259701394916$	$3299763765852494$	$174887471790752768$

Jacobians and polarizations

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

• $y^2=12x^6+24x^5+14x^4+5x^3+35x^2+35x+37$
• $y^2=25x^6+26x^5+28x^4+8x^3+40x^2+19x+30$
• $y^2=44x^6+17x^5+21x^4+20x^3+48x^2+36x+47$
• $y^2=x^6+8x^5+4x^4+25x^3+4x^2+48x+51$
• $y^2=39x^6+19x^5+27x^4+13x^3+27x^2+8x+30$
• $y^2=31x^6+7x^5+52x^4+10x^3+43x^2+52x+14$

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 4.0.2488896.4.

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_in	$2$	(not in LMFDB)