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

• Label: 2.53.ax_iv
• Base field: $\F_{53}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{53}$
Dimension:	$2$
L-polynomial:	$1 - 23 x + 229 x^{2} - 1219 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.0161938946932$, $\pm0.302684961462$
Angle rank:	$2$ (numerical)
Number field:	4.0.105413.1
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	2

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1797$	$7692957$	$22160670489$	$62251231105989$	$174870196450697472$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$31$	$2739$	$148855$	$7889411$	$418154186$	$22163827707$	$1174707089699$	$62259672066019$	$3299763563132611$	$174887470468463334$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=14x^6+6x^5+2x^4+7x^3+31x^2+50x+11$
• $y^2=12x^6+9x^5+35x^4+24x^3+23x^2+17x+27$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The endomorphism algebra of this simple isogeny class is 4.0.105413.1.

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.x_iv	$2$	(not in LMFDB)