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

• Label: 2.53.r_gq
• 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:	$( 1 + 6 x + 53 x^{2} )( 1 + 11 x + 53 x^{2} )$
	$1 + 17 x + 172 x^{2} + 901 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.635198170427$, $\pm0.772597778064$
Angle rank:	$2$ (numerical)
Jacobians:	$40$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3900$	$8049600$	$21992864400$	$62307928800000$	$174886481712247500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$71$	$2865$	$147722$	$7896593$	$418193131$	$22164237270$	$1174711393087$	$62259692373313$	$3299763633125666$	$174887469445302825$

Jacobians and polarizations

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

• $y^2=43 x^6+45 x^5+34 x^4+22 x^3+25 x^2+38 x+15$
• $y^2=27 x^6+11 x^5+30 x^4+2 x^3+24 x^2+15 x+44$
• $y^2=44 x^6+21 x^5+9 x^4+44 x^3+51 x^2+18 x+24$
• $y^2=8 x^6+9 x^5+41 x^4+31 x^3+7 x^2+12 x+49$
• $y^2=48 x^6+36 x^5+11 x^4+43 x^3+40 x^2+19 x+52$
• $y^2=3 x^6+24 x^5+34 x^4+7 x^3+22 x^2+35 x+24$
• $y^2=45 x^6+39 x^5+52 x^4+10 x^3+27 x^2+20 x+35$
• $y^2=43 x^6+25 x^5+2 x^4+20 x^3+21 x^2+52 x+40$
• $y^2=17 x^6+41 x^5+16 x^4+4 x^3+52 x^2+9 x+40$
• $y^2=29 x^6+21 x^5+30 x^4+35 x^3+39 x^2+35 x+14$
• $y^2=10 x^6+50 x^5+26 x^4+10 x^3+9 x^2+30 x+36$
• $y^2=19 x^6+13 x^5+11 x^4+12 x^3+30 x^2+47 x+18$
• $y^2=49 x^6+52 x^5+21 x^4+39 x^3+21 x^2+16 x+46$
• $y^2=21 x^6+2 x^5+21 x^4+7 x^3+34 x^2+48 x+23$
• $y^2=16 x^6+45 x^5+38 x^4+15 x^3+x^2+23 x+16$
• $y^2=44 x^6+50 x^5+23 x^4+7 x^3+49 x^2+12 x+28$
• $y^2=50 x^6+14 x^5+10 x^4+14 x^3+46 x^2+28 x+25$
• $y^2=16 x^6+23 x^5+37 x^4+42 x^3+x^2+x+52$
• $y^2=27 x^6+11 x^5+30 x^4+33 x^3+14 x^2+9 x+17$
• $y^2=11 x^6+25 x^5+17 x^4+42 x^3+29 x^2+11 x+13$
• and

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.g $\times$ 1.53.l and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.53.g : \(\Q(\sqrt{-11}) \).
• 1.53.l : \(\Q(\sqrt{-91}) \).

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_gq	$2$	(not in LMFDB)
2.53.af_bo	$2$	(not in LMFDB)
2.53.f_bo	$2$	(not in LMFDB)