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

• Label: 2.53.u_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:	$\pm0.740986412023$, $\pm0.740986412023$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4096$	$7929856$	$21989330944$	$62347826692096$	$174866938405851136$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$74$	$2822$	$147698$	$7901646$	$418146394$	$22164260438$	$1174714749058$	$62259659655838$	$3299763708068714$	$174887470832890022$

Jacobians and polarizations

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

• $y^2=x^6+17 x^5+20 x^4+28 x^3+44 x^2+42 x+13$
• $y^2=x^6+11 x^5+43 x^4+4 x^3+43 x^2+11 x+1$
• $y^2=28 x^6+26 x^5+x^4+40 x^3+x^2+26 x+28$
• $y^2=13 x^6+48 x^5+46 x^4+52 x^3+18 x^2+40 x+11$
• $y^2=6 x^6+39 x^4+39 x^2+6$
• $y^2=12 x^6+46 x^5+13 x^4+21 x^3+29 x^2+42 x+31$
• $y^2=24 x^6+17 x^5+37 x^3+18 x^2+33 x+17$
• $y^2=25 x^6+42 x^5+31 x^4+6 x^3+31 x^2+42 x+25$
• $y^2=48 x^6+39 x^5+39 x^4+46 x^3+20 x^2+15 x+47$
• $y^2=39 x^6+19 x^5+29 x^4+39 x^3+29 x^2+33 x+34$
• $y^2=42 x^6+10 x^4+10 x^2+42$
• $y^2=9 x^6+x^5+20 x^4+7 x^3+20 x^2+x+9$
• $y^2=47 x^6+17 x^5+16 x^4+39 x^3+46 x^2+37 x+10$
• $y^2=39 x^6+33 x^4+33 x^2+39$
• $y^2=44 x^6+46 x^5+51 x^4+47 x^3+51 x^2+46 x+44$
• $y^2=46 x^6+43 x^5+40 x^4+31 x^3+40 x^2+43 x+46$
• $y^2=22 x^6+17 x^5+42 x^4+7 x^3+42 x^2+17 x+22$
• $y^2=26 x^6+38 x^5+24 x^4+39 x^3+24 x^2+38 x+26$
• $y^2=29 x^6+22 x^4+22 x^2+29$
• $y^2=14 x^6+41 x^4+9 x^3+26 x^2+9 x+7$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.k^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.au_hy	$2$	(not in LMFDB)
2.53.a_g	$2$	(not in LMFDB)
2.53.ak_bv	$3$	(not in LMFDB)