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

• Label: 2.53.ax_ix
• 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:	$1 - 23 x + 231 x^{2} - 1219 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.0716818252882$, $\pm0.293214705465$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-58 +10 \sqrt{29}})\)
Galois group:	$C_4$
Jacobians:	$7$
Isomorphism classes:	7
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1799$	$7705117$	$22181347979$	$62270112738869$	$174881930871427024$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$31$	$2743$	$148993$	$7891803$	$418182246$	$22164079663$	$1174709013649$	$62259686667891$	$3299763690957319$	$174887471709640518$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which all are hyperelliptic):

• $y^2=41 x^6+8 x^5+32 x^4+18 x^3+24 x^2+40 x+49$
• $y^2=7 x^6+19 x^5+51 x^4+35 x^3+31 x^2+13 x+23$
• $y^2=33 x^6+27 x^5+51 x^4+15 x^3+20 x^2+37 x+33$
• $y^2=28 x^6+38 x^5+35 x^4+52 x^2+40 x+5$
• $y^2=19 x^6+27 x^5+45 x^4+x^3+35 x^2+28 x+41$
• $y^2=18 x^6+45 x^5+7 x^4+27 x^3+29 x^2+16 x+39$
• $y^2=29 x^6+34 x^5+52 x^4+31 x^3+29 x^2+11 x+24$

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 \(\Q(\sqrt{-58 +10 \sqrt{29}})\).

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_ix	$2$	(not in LMFDB)