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

• Label: 2.53.ax_jc
• 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:	$( 1 - 13 x + 53 x^{2} )( 1 - 10 x + 53 x^{2} )$
	$1 - 23 x + 236 x^{2} - 1219 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.148706751109$, $\pm0.259013587977$
Angle rank:	$2$ (numerical)
Jacobians:	$8$
Isomorphism classes:	20

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1804$	$7735552$	$22233066064$	$62316771472384$	$174909584820468124$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$31$	$2753$	$149338$	$7897713$	$418248371$	$22164591638$	$1174711484999$	$62259688095361$	$3299763589533874$	$174887470633173593$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=39x^6+35x^5+32x^4+7x^3+24x^2+34x+29$
• $y^2=39x^6+5x^5+29x^4+2x^3+20x^2+23x+27$
• $y^2=39x^6+17x^5+37x^4+36x^3+32x^2+28x+31$
• $y^2=45x^6+4x^5+3x^4+31x^3+33x^2+2x+34$
• $y^2=22x^6+11x^5+13x^4+19x^2+38x+8$
• $y^2=15x^6+13x^5+2x^4+50x^3+45x^2+39x+8$
• $y^2=31x^6+6x^5+41x^4+24x^3+49x^2+5x$
• $y^2=3x^6+14x^5+21x^4+16x^3+39x^2+47x+49$

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.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.an : \(\Q(\sqrt{-43}) \).
• 1.53.ak : \(\Q(\sqrt{-7}) \).

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.ad_ay	$2$	(not in LMFDB)
2.53.d_ay	$2$	(not in LMFDB)
2.53.x_jc	$2$	(not in LMFDB)