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

• Label: 2.53.k_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:	$\pm0.611579124397$, $\pm0.611579124397$
Angle rank:	$1$ (numerical)
Jacobians:	$17$
Cyclic group of points:	no
Non-cyclic primes:	$59$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3481$	$8357881$	$21965611264$	$62244825632521$	$174921116647446961$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$64$	$2972$	$147538$	$7888596$	$418275944$	$22164058838$	$1174708387448$	$62259720194788$	$3299763588761674$	$174887468802193772$

Jacobians and polarizations

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

• $y^2=38 x^6+49 x^5+19 x^4+50 x^3+40 x^2+44 x+9$
• $y^2=21 x^6+31 x^5+50 x^4+18 x^3+34 x^2+48 x+23$
• $y^2=22 x^6+21 x^5+13 x^4+15 x^3+17 x^2+45 x+47$
• $y^2=17 x^6+25 x^5+39 x^4+22 x^3+15 x^2+12 x+33$
• $y^2=50 x^6+13 x^5+26 x^4+48 x^3+51 x^2+16 x+37$
• $y^2=9 x^6+19 x^5+32 x^4+7 x^3+15 x^2+16 x+24$
• $y^2=35 x^6+18 x^5+2 x^4+15 x^3+6 x^2+17 x+4$
• $y^2=34 x^6+11 x^5+45 x^4+38 x^3+47 x+47$
• $y^2=4 x^6+19 x^5+50 x^4+20 x^3+45 x^2+35 x+15$
• $y^2=24 x^6+4 x^5+45 x^4+14 x^3+49 x^2+20 x+30$
• $y^2=26 x^6+49 x^5+27 x^4+9 x^3+29 x^2+5 x+42$
• $y^2=24 x^6+25 x^5+19 x^4+9 x^3+8 x^2+52 x+43$
• $y^2=18 x^6+24 x^5+25 x^4+50 x^3+37 x^2+12 x+50$
• $y^2=39 x^6+14 x^5+5 x^4+14 x^3+51 x^2+51 x+31$
• $y^2=23 x^6+37 x^5+48 x^4+2 x^3+25 x^2+20 x+22$
• $y^2=24 x^6+3 x^5+44 x^4+5 x^3+9 x^2+3 x+29$
• $y^2=36 x^6+21 x^5+39 x^4+48 x^3+x^2+11 x+11$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.f^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.ak_fb	$2$	(not in LMFDB)
2.53.a_dd	$2$	(not in LMFDB)
2.53.af_abc	$3$	(not in LMFDB)