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

• Label: 2.53.aq_go
• 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:	$( 1 - 8 x + 53 x^{2} )^{2}$
	$1 - 16 x + 170 x^{2} - 848 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.314840234458$, $\pm0.314840234458$
Angle rank:	$1$ (numerical)
Jacobians:	$13$
Cyclic group of points:	no
Non-cyclic primes:	$2, 23$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2116$	$8133904$	$22391531044$	$62320540880896$	$174879569069114116$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$38$	$2894$	$150398$	$7898190$	$418176598$	$22163801438$	$1174707663790$	$62259692266654$	$3299763790875014$	$174887471859765614$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 13 curves (of which all are hyperelliptic):

• $y^2=30 x^6+30 x^5+44 x^4+44 x^3+23 x^2+7 x+32$
• $y^2=37 x^6+47 x^5+23 x^4+44 x^3+23 x^2+47 x+37$
• $y^2=3 x^6+2 x^5+30 x^4+21 x^3+28 x^2+x+3$
• $y^2=13 x^6+41 x^5+44 x^4+47 x^3+32 x^2+3 x+35$
• $y^2=22 x^6+x^5+3 x^4+33 x^3+51 x^2+42 x+19$
• $y^2=32 x^6+7 x^4+7 x^2+32$
• $y^2=13 x^6+16 x^5+3 x^4+24 x^3+19 x^2+33 x+17$
• $y^2=25 x^6+36 x^5+4 x^4+22 x^3+18 x^2+30 x+8$
• $y^2=33 x^6+25 x^5+24 x^4+24 x^3+37 x^2+28 x+36$
• $y^2=47 x^6+37 x^5+19 x^4+17 x^3+19 x^2+37 x+47$
• $y^2=35 x^6+16 x^5+39 x^4+38 x^3+42 x^2+2 x+14$
• $y^2=40 x^6+9 x^5+29 x^4+16 x^3+35 x^2+39 x+25$
• $y^2=45 x^6+x^5+26 x^4+21 x^3+28 x^2+12 x+39$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.ai^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-37}) \)$)$

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_bq	$2$	(not in LMFDB)
2.53.q_go	$2$	(not in LMFDB)
2.53.i_l	$3$	(not in LMFDB)