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

• Label: 2.53.ai_ej
• 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:	$( 1 - 7 x + 53 x^{2} )( 1 - x + 53 x^{2} )$
	$1 - 8 x + 113 x^{2} - 424 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.340360113580$, $\pm0.478121163875$
Angle rank:	$2$ (numerical)
Jacobians:	$33$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2491$	$8357305$	$22302939328$	$62235722098825$	$174871576761440371$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$46$	$2972$	$149806$	$7887444$	$418157486$	$22164338774$	$1174711361030$	$62259687125476$	$3299763637805878$	$174887471261431532$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 33 curves (of which all are hyperelliptic):

• $y^2=27 x^6+30 x^5+10 x^4+9 x^3+10 x+6$
• $y^2=10 x^6+35 x^5+35 x^4+42 x^3+35 x^2+35 x+10$
• $y^2=6 x^6+52 x^5+7 x^4+5 x^3+7 x^2+52 x+6$
• $y^2=45 x^6+34 x^5+48 x^4+24 x^3+7 x^2+14 x+1$
• $y^2=5 x^6+3 x^5+5 x^4+2 x^3+14 x^2+27 x+5$
• $y^2=x^6+12 x^5+26 x^4+19 x^3+19 x^2+36 x+34$
• $y^2=23 x^6+46 x^5+50 x^4+7 x^3+37 x^2+3 x+41$
• $y^2=23 x^6+18 x^5+39 x^4+20 x^3+39 x^2+18 x+23$
• $y^2=28 x^6+38 x^5+9 x^4+15 x^3+41 x^2+x+50$
• $y^2=42 x^6+13 x^5+15 x^4+15 x^3+50 x^2+20 x+46$
• $y^2=14 x^6+8 x^5+x^4+16 x^3+3 x^2+23 x+7$
• $y^2=25 x^6+41 x^5+25 x^4+15 x^3+25 x^2+41 x+25$
• $y^2=47 x^6+44 x^5+27 x^4+17 x^3+42 x^2+11 x+39$
• $y^2=49 x^6+45 x^5+15 x^4+44 x^3+15 x^2+45 x+49$
• $y^2=37 x^6+22 x^5+10 x^4+39 x^3+12 x^2+39 x+45$
• $y^2=5 x^6+45 x^5+5 x^4+23 x^3+5 x^2+45 x+5$
• $y^2=12 x^6+28 x^5+25 x^4+48 x^3+28 x^2+3 x+51$
• $y^2=20 x^6+34 x^5+12 x^4+20 x^3+26 x^2+27 x+21$
• $y^2=51 x^6+43 x^5+30 x^4+40 x^3+22 x^2+37 x+3$
• $y^2=42 x^6+47 x^5+42 x^4+52 x^3+10 x^2+20 x+39$
• and

• $y^2=23 x^6+47 x^5+11 x^4+10 x^3+51 x^2+35 x+26$
• $y^2=8 x^6+3 x^5+43 x^4+9 x^3+43 x^2+3 x+8$
• $y^2=12 x^6+14 x^5+15 x^4+37 x^3+45 x^2+33 x+45$
• $y^2=12 x^6+40 x^5+5 x^4+16 x^3+25 x^2+40 x+7$
• $y^2=20 x^6+41 x^5+21 x^4+8 x^3+21 x^2+41 x+20$
• $y^2=8 x^6+7 x^5+48 x^4+46 x^3+27 x^2+4 x+45$
• $y^2=2 x^6+15 x^5+46 x^4+38 x^3+46 x^2+15 x+2$
• $y^2=22 x^6+21 x^5+33 x^4+3 x^3+38 x^2+12 x+7$
• $y^2=6 x^6+30 x^5+23 x^4+38 x^3+51 x^2+22 x+8$
• $y^2=3 x^6+38 x^5+40 x^4+46 x^3+5 x^2+8 x+38$
• $y^2=35 x^6+27 x^5+36 x^4+27 x^3+9 x^2+8 x+16$
• $y^2=18 x^6+42 x^4+7 x^3+30 x^2+52 x+36$
• $y^2=45 x^6+28 x^5+15 x^4+32 x^3+39 x^2+13 x+35$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.ah $\times$ 1.53.ab 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.ah : \(\Q(\sqrt{-163}) \).
• 1.53.ab : \(\Q(\sqrt{-211}) \).

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.ag_dv	$2$	(not in LMFDB)
2.53.g_dv	$2$	(not in LMFDB)
2.53.i_ej	$2$	(not in LMFDB)