Abelian variety isogeny class 2.59.ax_jn over $\F_{59}$

• Label: 2.59.ax_jn
• Base field: $\F_{59}$
• 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:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$1 - 23 x + 247 x^{2} - 1357 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.166722977967$, $\pm0.282548125847$
Angle rank:	$2$ (numerical)
Number field:	4.0.4901.1
Galois group:	$D_{4}$
Jacobians:	$17$
Isomorphism classes:	17

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2349$	$12001041$	$42346275687$	$146950118817021$	$511160231377803024$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$3447$	$206185$	$12127235$	$714985112$	$42180707907$	$2488651067807$	$146830433123539$	$8662995843390175$	$511116753803802102$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 17 curves (of which all are hyperelliptic):

• $y^2=39 x^6+54 x^5+9 x^4+22 x^3+42 x^2+20 x+39$
• $y^2=13 x^6+53 x^5+22 x^4+43 x^3+10 x^2+25 x+12$
• $y^2=25 x^6+41 x^5+53 x^4+35 x^3+54 x^2+43 x+15$
• $y^2=11 x^6+6 x^5+37 x^4+44 x^3+39 x^2+51 x+2$
• $y^2=19 x^6+41 x^5+18 x^4+18 x^3+34 x^2+46 x+31$
• $y^2=24 x^6+37 x^5+41 x^4+31 x^3+35 x^2+7 x+33$
• $y^2=39 x^6+51 x^5+2 x^4+36 x^3+30 x^2+13 x+40$
• $y^2=57 x^6+9 x^5+26 x^4+22 x^3+53 x^2+50 x+2$
• $y^2=19 x^6+44 x^5+54 x^4+35 x^3+21 x^2+22 x+41$
• $y^2=2 x^6+14 x^5+8 x^4+45 x^3+39 x^2+9 x+14$
• $y^2=8 x^6+8 x^5+42 x^4+2 x^3+14 x^2+52 x+32$
• $y^2=37 x^6+7 x^5+39 x^4+33 x^3+32 x^2+41 x+20$
• $y^2=2 x^6+44 x^5+17 x^4+26 x^3+40 x^2+8 x+34$
• $y^2=44 x^6+19 x^5+52 x^4+23 x^3+40 x^2+46 x+13$
• $y^2=8 x^6+36 x^5+27 x^4+4 x^3+44 x^2+5 x+33$
• $y^2=43 x^6+30 x^5+43 x^4+45 x^2+51 x+56$
• $y^2=38 x^6+42 x^5+18 x^4+51 x^3+33 x^2+26 x+13$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{59}$.

Endomorphism algebra over $\F_{59}$

The endomorphism algebra of this simple isogeny class is 4.0.4901.1.

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
2.59.x_jn	$2$	(not in LMFDB)