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

• Label: 2.53.ak_fb
• 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:	$( 1 - 5 x + 53 x^{2} )^{2}$
	$1 - 10 x + 131 x^{2} - 530 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.388420875603$, $\pm0.388420875603$
Angle rank:	$1$ (numerical)
Jacobians:	$17$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2401$	$8357881$	$22364604304$	$62244825632521$	$174853828992462361$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$44$	$2972$	$150218$	$7888596$	$418115044$	$22164058838$	$1174713892228$	$62259720194788$	$3299763594842594$	$174887468802193772$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 17 curves (of which all are hyperelliptic):

• $y^2=23 x^6+45 x^5+38 x^4+47 x^3+27 x^2+35 x+18$
• $y^2=42 x^6+9 x^5+47 x^4+36 x^3+15 x^2+43 x+46$
• $y^2=11 x^6+37 x^5+33 x^4+34 x^3+35 x^2+49 x+50$
• $y^2=34 x^6+50 x^5+25 x^4+44 x^3+30 x^2+24 x+13$
• $y^2=25 x^6+33 x^5+13 x^4+24 x^3+52 x^2+8 x+45$
• $y^2=18 x^6+38 x^5+11 x^4+14 x^3+30 x^2+32 x+48$
• $y^2=17 x^6+36 x^5+4 x^4+30 x^3+12 x^2+34 x+8$
• $y^2=17 x^6+32 x^5+49 x^4+19 x^3+50 x+50$
• $y^2=8 x^6+38 x^5+47 x^4+40 x^3+37 x^2+17 x+30$
• $y^2=48 x^6+8 x^5+37 x^4+28 x^3+45 x^2+40 x+7$
• $y^2=52 x^6+45 x^5+x^4+18 x^3+5 x^2+10 x+31$
• $y^2=12 x^6+39 x^5+36 x^4+31 x^3+4 x^2+26 x+48$
• $y^2=36 x^6+48 x^5+50 x^4+47 x^3+21 x^2+24 x+47$
• $y^2=25 x^6+28 x^5+10 x^4+28 x^3+49 x^2+49 x+9$
• $y^2=38 x^6+45 x^5+24 x^4+x^3+39 x^2+10 x+11$
• $y^2=48 x^6+6 x^5+35 x^4+10 x^3+18 x^2+6 x+5$
• $y^2=18 x^6+37 x^5+46 x^4+24 x^3+27 x^2+32 x+32$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.af^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-187}) \)$)$

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_dd	$2$	(not in LMFDB)
2.53.k_fb	$2$	(not in LMFDB)
2.53.f_abc	$3$	(not in LMFDB)