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

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

Invariants

Base field:	$\F_{53}$
Dimension:	$2$
L-polynomial:	$( 1 - 10 x + 53 x^{2} )^{2}$
	$1 - 20 x + 206 x^{2} - 1060 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.259013587977$, $\pm0.259013587977$
Angle rank:	$1$ (numerical)
Jacobians:	$20$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1936$	$7929856$	$22340683024$	$62347826692096$	$174908005203361936$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$34$	$2822$	$150058$	$7901646$	$418244594$	$22164260438$	$1174707530618$	$62259659655838$	$3299763475535554$	$174887470832890022$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

• $y^2=27 x^6+35 x^5+10 x^4+14 x^3+22 x^2+21 x+33$
• $y^2=27 x^6+32 x^5+48 x^4+2 x^3+48 x^2+32 x+27$
• $y^2=3 x^6+52 x^5+2 x^4+27 x^3+2 x^2+52 x+3$
• $y^2=33 x^6+24 x^5+23 x^4+26 x^3+9 x^2+20 x+32$
• $y^2=3 x^6+46 x^4+46 x^2+3$
• $y^2=6 x^6+23 x^5+33 x^4+37 x^3+41 x^2+21 x+42$
• $y^2=12 x^6+35 x^5+45 x^3+9 x^2+43 x+35$
• $y^2=50 x^6+31 x^5+9 x^4+12 x^3+9 x^2+31 x+50$
• $y^2=24 x^6+46 x^5+46 x^4+23 x^3+10 x^2+34 x+50$
• $y^2=25 x^6+38 x^5+5 x^4+25 x^3+5 x^2+13 x+15$
• $y^2=31 x^6+20 x^4+20 x^2+31$
• $y^2=31 x^6+27 x^5+10 x^4+30 x^3+10 x^2+27 x+31$
• $y^2=41 x^6+34 x^5+32 x^4+25 x^3+39 x^2+21 x+20$
• $y^2=46 x^6+43 x^4+43 x^2+46$
• $y^2=35 x^6+39 x^5+49 x^4+41 x^3+49 x^2+39 x+35$
• $y^2=39 x^6+33 x^5+27 x^4+9 x^3+27 x^2+33 x+39$
• $y^2=44 x^6+34 x^5+31 x^4+14 x^3+31 x^2+34 x+44$
• $y^2=13 x^6+19 x^5+12 x^4+46 x^3+12 x^2+19 x+13$
• $y^2=5 x^6+44 x^4+44 x^2+5$
• $y^2=28 x^6+29 x^4+18 x^3+52 x^2+18 x+14$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.ak^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.53.a_g	$2$	(not in LMFDB)
2.53.u_hy	$2$	(not in LMFDB)
2.53.k_bv	$3$	(not in LMFDB)