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

• Label: 2.53.av_ic
• 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:	$( 1 - 13 x + 53 x^{2} )( 1 - 8 x + 53 x^{2} )$
	$1 - 21 x + 210 x^{2} - 1113 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.148706751109$, $\pm0.314840234458$
Angle rank:	$2$ (numerical)
Jacobians:	$6$
Isomorphism classes:	26
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1886$	$7834444$	$22258353224$	$62303133870016$	$174895366047001766$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$33$	$2789$	$149508$	$7895985$	$418214373$	$22164362138$	$1174711551585$	$62259704400769$	$3299763747203604$	$174887471146611389$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):

• $y^2=2 x^6+21 x^5+x^4+5 x^3+27 x^2+12 x+33$
• $y^2=29 x^6+19 x^5+45 x^4+30 x^3+19 x^2+20 x+29$
• $y^2=39 x^6+x^5+30 x^4+31 x^3+15 x^2+32 x+33$
• $y^2=3 x^6+2 x^5+48 x^4+38 x^3+24 x^2+3 x+34$
• $y^2=26 x^6+31 x^5+33 x^4+41 x^3+47 x^2+17 x+20$
• $y^2=48 x^6+40 x^5+15 x^4+23 x^3+12 x^2+11 x+7$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.an $\times$ 1.53.ai 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.an : \(\Q(\sqrt{-43}) \).
• 1.53.ai : \(\Q(\sqrt{-37}) \).

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.af_c	$2$	(not in LMFDB)
2.53.f_c	$2$	(not in LMFDB)
2.53.v_ic	$2$	(not in LMFDB)