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

• Label: 2.53.s_hf
• 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:	$( 1 + 9 x + 53 x^{2} )^{2}$
	$1 + 18 x + 187 x^{2} + 954 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.712106452697$, $\pm0.712106452697$
Angle rank:	$1$ (numerical)
Jacobians:	$40$
Cyclic group of points:	no
Non-cyclic primes:	$3, 7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3969$	$8037225$	$21956126976$	$62338525475625$	$174881005122944169$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$72$	$2860$	$147474$	$7900468$	$418180032$	$22163971030$	$1174715470224$	$62259672113188$	$3299763526975242$	$174887471918758300$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 40 curves (of which all are hyperelliptic):

• $y^2=36 x^6+41 x^5+39 x^4+47 x^3+21 x^2+26 x+11$
• $y^2=46 x^6+5 x^5+39 x^4+30 x^3+22 x^2+21 x+25$
• $y^2=38 x^6+3 x^5+43 x^4+12 x^3+38 x^2+20 x+9$
• $y^2=43 x^6+26 x^5+7 x^4+9 x^3+9 x^2+30 x+37$
• $y^2=37 x^6+40 x^5+18 x^4+2 x^3+5 x^2+43 x+7$
• $y^2=41 x^6+9 x^5+52 x^4+20 x^3+15 x^2+11 x+8$
• $y^2=30 x^6+3 x^5+18 x^4+23 x^3+26 x^2+x+12$
• $y^2=11 x^6+36 x^5+33 x^4+31 x^3+14 x^2+24 x+9$
• $y^2=52 x^6+47 x^5+15 x^4+41 x^3+38 x^2+47 x+1$
• $y^2=52 x^6+38 x^5+34 x^4+23 x^3+30 x^2+25 x+42$
• $y^2=49 x^6+35 x^5+37 x^4+45 x^3+25 x^2+31 x+15$
• $y^2=44 x^6+24 x^5+45 x^4+31 x^3+36 x^2+8 x+6$
• $y^2=14 x^6+46 x^5+46 x^4+52 x^3+15 x^2+36 x+18$
• $y^2=30 x^6+5 x^5+44 x^4+22 x^3+29 x^2+12 x+33$
• $y^2=16 x^6+x^5+46 x^4+33 x^3+25 x^2+16 x+17$
• $y^2=10 x^6+9 x^5+11 x^4+x^3+42 x^2+9 x+43$
• $y^2=7 x^6+23 x^5+21 x^4+47 x^3+35 x^2+5 x+1$
• $y^2=52 x^6+23 x^4+9 x^3+22 x^2+9$
• $y^2=13 x^6+32 x^5+44 x^4+22 x^3+6 x^2+26 x+4$
• $y^2=4 x^6+48 x^5+19 x^4+46 x^3+19 x^2+48 x+4$
• and

• $y^2=14 x^6+33 x^5+4 x^4+12 x^3+42 x^2+31 x+30$
• $y^2=39 x^6+25 x^5+43 x^4+15 x^2+43 x+34$
• $y^2=48 x^6+4 x^5+33 x^4+40 x^3+2 x^2+17 x+22$
• $y^2=29 x^6+22 x^5+8 x^4+9 x^3+5 x^2+50 x+43$
• $y^2=47 x^6+49 x^5+22 x^4+21 x^3+51 x^2+28 x+1$
• $y^2=52 x^6+18 x^5+37 x^4+25 x^3+46 x^2+8 x+16$
• $y^2=4 x^6+25 x^5+23 x^4+5 x^3+15 x^2+34 x+9$
• $y^2=44 x^6+x^5+48 x^4+33 x^3+22 x^2+13 x+6$
• $y^2=9 x^6+51 x^5+26 x^4+19 x^3+2 x^2+21 x+7$
• $y^2=27 x^6+36 x^5+4 x^4+38 x^3+13 x^2+49 x+30$
• $y^2=52 x^6+48 x^5+3 x^4+8 x^3+39 x^2+3 x+29$
• $y^2=18 x^6+45 x^5+17 x^4+4 x^3+15 x^2+31 x+39$
• $y^2=5 x^6+31 x^5+18 x^4+52 x^3+22 x^2+45 x+14$
• $y^2=48 x^6+5 x^5+25 x^4+39 x^3+28 x^2+5 x+5$
• $y^2=40 x^6+5 x^5+32 x^4+15 x^3+51 x^2+23 x+15$
• $y^2=49 x^6+20 x^5+25 x^4+51 x^3+28 x^2+20 x+4$
• $y^2=x^6+2 x^5+26 x^4+51 x^3+21 x^2+30 x+6$
• $y^2=36 x^6+36 x^5+26 x^4+21 x^3+26 x^2+36 x+36$
• $y^2=9 x^6+46 x^5+19 x^4+11 x^3+39 x^2+10 x+29$
• $y^2=42 x^6+36 x^5+43 x^4+40 x^3+41 x^2+44 x+44$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.j^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-131}) \)$)$

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.as_hf	$2$	(not in LMFDB)
2.53.a_z	$2$	(not in LMFDB)
2.53.aj_bc	$3$	(not in LMFDB)