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

• Label: 2.53.y_jp
• 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:	$( 1 + 11 x + 53 x^{2} )( 1 + 13 x + 53 x^{2} )$
	$1 + 24 x + 249 x^{2} + 1272 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.772597778064$, $\pm0.851293248891$
Angle rank:	$2$ (numerical)
Jacobians:	$6$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4355$	$7677865$	$22121727680$	$62315279859625$	$174860082122715275$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$78$	$2732$	$148590$	$7897524$	$418129998$	$22164765014$	$1174709606694$	$62259690169636$	$3299763649594230$	$174887469854341532$

Jacobians and polarizations

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

• $y^2=48 x^6+21 x^5+12 x^4+36 x^3+12 x^2+21 x+48$
• $y^2=49 x^6+16 x^5+5 x^4+40 x^3+5 x^2+16 x+49$
• $y^2=42 x^6+38 x^5+9 x^4+44 x^3+9 x^2+38 x+42$
• $y^2=51 x^6+49 x^5+12 x^4+21 x^3+12 x^2+49 x+51$
• $y^2=31 x^6+29 x^5+39 x^4+46 x^3+39 x^2+29 x+31$
• $y^2=46 x^6+14 x^5+42 x^4+23 x^3+42 x^2+14 x+46$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.l $\times$ 1.53.n 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.l : \(\Q(\sqrt{-91}) \).
• 1.53.n : \(\Q(\sqrt{-43}) \).

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.ay_jp	$2$	(not in LMFDB)
2.53.ac_abl	$2$	(not in LMFDB)
2.53.c_abl	$2$	(not in LMFDB)