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

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

Invariants

Base field:	$\F_{53}$
Dimension:	$2$
L-polynomial:	$1 + 17 x + 167 x^{2} + 901 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.614981926117$, $\pm0.802792878183$
Angle rank:	$2$ (numerical)
Number field:	4.0.331525.2
Galois group:	$D_{4}$
Jacobians:	$48$
Cyclic group of points:	yes

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3895$	$8019805$	$22030669195$	$62289063553525$	$174887014580098000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$71$	$2855$	$147977$	$7894203$	$418194406$	$22164443615$	$1174709097577$	$62259701882803$	$3299763666582431$	$174887468786894150$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 48 curves (of which all are hyperelliptic):

• $y^2=14 x^6+3 x^5+31 x^4+45 x^3+13 x^2+43 x+34$
• $y^2=38 x^6+15 x^5+12 x^4+33 x^3+21 x^2+9 x+5$
• $y^2=10 x^6+10 x^5+39 x^4+18 x^3+7 x^2+5 x+15$
• $y^2=5 x^6+30 x^5+x^4+41 x^3+17 x^2+4 x+38$
• $y^2=14 x^6+27 x^5+29 x^4+40 x^3+25 x^2+13 x+26$
• $y^2=52 x^6+13 x^5+24 x^4+26 x^3+47 x^2+40 x$
• $y^2=49 x^6+47 x^5+15 x^4+x^3+8 x^2+29 x+4$
• $y^2=17 x^6+48 x^5+25 x^4+4 x^3+46 x^2+35 x+28$
• $y^2=27 x^6+13 x^5+49 x^4+34 x^3+3 x^2+34 x+19$
• $y^2=22 x^6+19 x^5+32 x^4+41 x^3+14 x^2+12 x+33$
• $y^2=51 x^6+9 x^5+36 x^4+38 x^3+40 x^2+3 x+29$
• $y^2=2 x^6+24 x^5+51 x^4+19 x^3+16 x^2+26 x+10$
• $y^2=42 x^6+16 x^5+48 x^4+26 x^3+12 x^2+11 x+40$
• $y^2=15 x^6+34 x^5+37 x^4+3 x^3+28 x^2+24 x+12$
• $y^2=6 x^6+43 x^5+29 x^4+38 x^3+30 x^2+26 x+13$
• $y^2=38 x^6+19 x^5+21 x^4+37 x^3+39 x^2+41 x+41$
• $y^2=17 x^6+25 x^5+36 x^3+52 x^2+9 x+7$
• $y^2=12 x^6+51 x^5+7 x^4+14 x^3+31 x^2+4 x+29$
• $y^2=5 x^6+16 x^5+33 x^4+46 x^3+25 x^2+37 x+7$
• $y^2=17 x^6+7 x^5+3 x^4+46 x^3+25 x^2+47 x+2$
• and

• $y^2=x^6+20 x^5+33 x^4+23 x^3+10 x^2+50 x+28$
• $y^2=45 x^6+34 x^5+49 x^4+20 x^3+41 x^2+25 x+1$
• $y^2=51 x^6+31 x^5+35 x^4+11 x^3+39 x+47$
• $y^2=x^6+31 x^5+12 x^4+18 x^3+39 x^2+37 x+18$
• $y^2=8 x^6+47 x^5+51 x^4+5 x^3+18 x^2+52 x+25$
• $y^2=34 x^6+20 x^5+29 x^4+31 x^3+19 x^2+18 x+15$
• $y^2=33 x^6+13 x^5+21 x^3+43 x^2+28 x+37$
• $y^2=43 x^6+41 x^5+30 x^4+29 x^3+28 x^2+27 x+29$
• $y^2=46 x^6+3 x^5+40 x^4+29 x^3+33 x^2+41 x+50$
• $y^2=11 x^6+5 x^5+20 x^3+39 x^2+10 x+48$
• $y^2=41 x^6+34 x^5+22 x^4+2 x^3+2 x^2+20 x+20$
• $y^2=49 x^6+11 x^5+8 x^4+43 x^3+14 x^2+10 x+26$
• $y^2=52 x^6+51 x^5+3 x^4+45 x^3+8 x^2+x+44$
• $y^2=14 x^6+52 x^5+34 x^4+52 x^3+12 x^2+33 x$
• $y^2=49 x^6+10 x^5+36 x^4+47 x^3+16 x^2+41 x+8$
• $y^2=24 x^6+6 x^5+32 x^4+44 x^3+51 x^2+7 x+44$
• $y^2=44 x^6+10 x^5+46 x^4+3 x^3+42 x^2+18 x+33$
• $y^2=36 x^6+36 x^5+23 x^4+39 x^3+13 x^2+42 x+17$
• $y^2=10 x^6+7 x^5+34 x^4+14 x^3+37 x^2+10 x+46$
• $y^2=9 x^6+20 x^5+20 x^4+32 x^3+21 x^2+11 x+48$
• $y^2=17 x^6+29 x^5+16 x^4+50 x^3+13 x^2+43 x+16$
• $y^2=12 x^6+26 x^5+30 x^4+10 x^3+6 x^2+x+12$
• $y^2=45 x^6+25 x^5+52 x^4+50 x^3+43 x^2+19 x+47$
• $y^2=29 x^6+17 x^5+48 x^4+2 x^3+4 x^2+24 x+45$
• $y^2=37 x^6+16 x^5+41 x^4+22 x^3+39 x^2+9 x+47$
• $y^2=14 x^6+15 x^5+49 x^4+35 x^3+29 x^2+39 x+6$
• $y^2=26 x^6+x^5+45 x^4+46 x^3+7 x^2+16 x+39$
• $y^2=4 x^6+52 x^5+22 x^4+12 x^3+21 x^2+7 x+37$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The endomorphism algebra of this simple isogeny class is 4.0.331525.2.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.53.ar_gl	$2$	(not in LMFDB)