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

• Label: 2.53.ay_jq
• 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:	$( 1 - 12 x + 53 x^{2} )^{2}$
	$1 - 24 x + 250 x^{2} - 1272 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.191645762723$, $\pm0.191645762723$
Angle rank:	$1$ (numerical)
Jacobians:	$16$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1764$	$7683984$	$22218287364$	$62325593358336$	$174921386827825764$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$2734$	$149238$	$7898830$	$418276590$	$22164891838$	$1174713210246$	$62259687128734$	$3299763442678974$	$174887468750014414$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):

• $y^2=51 x^6+25 x^5+16 x^3+11 x+26$
• $y^2=30 x^6+25 x^5+9 x^4+46 x^3+10 x^2+42 x+30$
• $y^2=9 x^6+28 x^5+44 x^4+35 x^3+15 x^2+13 x+29$
• $y^2=22 x^6+16 x^5+35 x^4+15 x^3+8 x^2+47 x+39$
• $y^2=34 x^6+5 x^5+35 x^4+46 x^3+35 x^2+5 x+34$
• $y^2=14 x^6+5 x^5+26 x^4+4 x^3+33 x^2+50 x+35$
• $y^2=23 x^6+29 x^5+23 x^4+26 x^3+3 x^2+4 x+26$
• $y^2=42 x^6+18 x^4+18 x^2+42$
• $y^2=23 x^6+4 x^5+23 x^4+24 x^3+x^2+19 x+21$
• $y^2=5 x^6+23 x^5+23 x^4+8 x^3+23 x^2+23 x+5$
• $y^2=33 x^6+2 x^4+2 x^2+33$
• $y^2=50 x^6+46 x^5+36 x^4+14 x^3+13 x^2+28 x+22$
• $y^2=51 x^6+48 x^5+35 x^4+19 x^3+35 x^2+48 x+51$
• $y^2=15 x^6+33 x^5+44 x^4+34 x^3+46 x^2+18 x+16$
• $y^2=14 x^6+5 x^5+13 x^4+43 x^3+42 x^2+34 x+23$
• $y^2=18 x^6+27 x^5+34 x^4+48 x^3+34 x^2+27 x+18$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.am^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-17}) \)$)$

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.a_abm	$2$	(not in LMFDB)
2.53.y_jq	$2$	(not in LMFDB)
2.53.m_dn	$3$	(not in LMFDB)