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

• Label: 2.53.ay_jm
• 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:	$( 1 - 14 x + 53 x^{2} )( 1 - 10 x + 53 x^{2} )$
	$1 - 24 x + 246 x^{2} - 1272 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.0885855327829$, $\pm0.259013587977$
Angle rank:	$2$ (numerical)
Jacobians:	$14$
Isomorphism classes:	72

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1760$	$7659520$	$22175072480$	$62284152832000$	$174894687404280800$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$2726$	$148950$	$7893582$	$418212750$	$22164340214$	$1174710133830$	$62259684654238$	$3299763626031870$	$174887471382390086$

Jacobians and polarizations

This isogeny class contains the Jacobians of 14 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=35x^6+13x^5+5x^4+9x^3+5x^2+13x+35$
• $y^2=2x^6+45x^5+37x^4+7x^3+37x^2+45x+2$
• $y^2=x^6+32x^5+24x^4+7x^3+24x^2+32x+1$
• $y^2=30x^6+6x^5+14x^4+28x^3+14x^2+6x+30$
• $y^2=24x^6+5x^5+26x^4+50x^3+5x^2+33x+50$
• $y^2=39x^6+15x^5+x^4+46x^3+9x^2+49x+23$
• $y^2=51x^6+46x^5+19x^4+6x^2+14x+40$
• $y^2=5x^6+47x^5+42x^4+26x^3+36x^2+x+22$
• $y^2=33x^6+48x^5+44x^4+49x^3+44x^2+48x+33$
• $y^2=46x^6+32x^5+48x^4+51x^3+48x^2+32x+46$
• $y^2=5x^6+36x^5+33x^4+27x^3+45x^2+10x+30$
• $y^2=46x^6+42x^5+51x^4+6x^3+30x^2+16x+40$
• $y^2=48x^6+48x^5+3x^4+30x^3+x^2+28x+21$
• $y^2=24x^6+10x^5+41x^4+26x^3+45x^2+2x+44$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.ao $\times$ 1.53.ak 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.ao : \(\Q(\sqrt{-1}) \).
• 1.53.ak : \(\Q(\sqrt{-7}) \).

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.ae_abi	$2$	(not in LMFDB)
2.53.e_abi	$2$	(not in LMFDB)
2.53.y_jm	$2$	(not in LMFDB)