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

• Label: 2.59.s_ho
• 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:	$1 + 18 x + 196 x^{2} + 1062 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.656866227456$, $\pm0.746191941696$
Angle rank:	$2$ (numerical)
Number field:	4.0.3187008.2
Galois group:	$D_{4}$
Jacobians:	$40$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4758$	$12361284$	$41859918126$	$146947631367888$	$511115350859437038$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$78$	$3550$	$203814$	$12127030$	$714922338$	$42180100414$	$2488654851306$	$146830430349598$	$8662995756326574$	$511116753777336430$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 40 curves (of which all are hyperelliptic):

• $y^2=28 x^6+23 x^5+56 x^4+38 x^3+16 x^2+57 x+8$
• $y^2=5 x^6+35 x^5+7 x^4+42 x^3+53 x^2+16 x+21$
• $y^2=52 x^6+31 x^5+58 x^4+34 x^3+11 x^2+15 x+46$
• $y^2=53 x^5+57 x^4+2 x^3+23 x^2+24 x+15$
• $y^2=11 x^6+12 x^5+x^4+46 x^3+23 x^2+55 x+5$
• $y^2=x^6+17 x^5+36 x^4+35 x^3+50 x^2+37 x+42$
• $y^2=28 x^6+26 x^5+24 x^4+36 x^3+22 x^2+28 x+10$
• $y^2=33 x^6+15 x^5+17 x^4+23 x^3+3 x^2+38 x+17$
• $y^2=28 x^6+15 x^5+58 x^4+6 x^3+31 x^2+29 x+41$
• $y^2=57 x^6+43 x^5+34 x^4+x^3+47 x^2+50 x+37$
• $y^2=36 x^6+41 x^5+17 x^4+17 x^3+27 x^2+38 x+57$
• $y^2=46 x^6+40 x^5+41 x^4+7 x^3+31 x^2+2 x+21$
• $y^2=21 x^6+29 x^5+48 x^4+12 x^3+48 x^2+58 x+42$
• $y^2=17 x^6+41 x^5+12 x^4+51 x^3+47 x^2+38 x+19$
• $y^2=19 x^6+2 x^5+56 x^4+19 x^3+15 x^2+53 x+20$
• $y^2=19 x^6+48 x^5+45 x^4+26 x^3+12 x^2+31 x+16$
• $y^2=15 x^6+19 x^5+26 x^4+30 x^3+7 x^2+44 x+45$
• $y^2=41 x^6+57 x^5+25 x^4+47 x^3+52 x^2+22 x+15$
• $y^2=14 x^6+16 x^5+10 x^3+8 x^2+8 x+16$
• $y^2=53 x^6+40 x^5+47 x^4+20 x^3+25 x^2+50 x+7$
• and

• $y^2=34 x^6+45 x^5+33 x^4+21 x^3+11 x^2+5 x+44$
• $y^2=43 x^6+39 x^5+41 x^4+55 x^3+11 x^2+26 x+40$
• $y^2=22 x^6+7 x^5+34 x^4+40 x^3+42 x^2+20 x+53$
• $y^2=19 x^6+33 x^5+7 x^4+6 x^3+4 x^2+42 x+9$
• $y^2=56 x^6+20 x^5+58 x^4+10 x^3+28 x^2+42 x+16$
• $y^2=27 x^6+46 x^5+9 x^4+37 x^3+46 x^2+36 x+14$
• $y^2=35 x^6+28 x^5+50 x^4+24 x^3+28 x^2+33 x+4$
• $y^2=16 x^6+2 x^5+39 x^4+19 x^3+16 x^2+29 x+37$
• $y^2=57 x^6+38 x^5+2 x^4+41 x^3+5 x^2+48 x+54$
• $y^2=37 x^6+20 x^5+32 x^4+24 x^3+3 x^2+32 x+19$
• $y^2=43 x^6+12 x^5+38 x^4+13 x^3+50 x^2+51 x+55$
• $y^2=19 x^6+11 x^5+43 x^4+32 x^3+42 x^2+54 x+57$
• $y^2=x^6+36 x^5+7 x^4+53 x^3+5 x^2+x+42$
• $y^2=44 x^6+20 x^5+54 x^4+4 x^3+3 x^2+58 x+54$
• $y^2=26 x^6+39 x^5+40 x^4+22 x^3+26 x^2+14 x+38$
• $y^2=49 x^6+28 x^5+24 x^4+51 x^2+35 x+41$
• $y^2=53 x^6+20 x^5+32 x^4+55 x^3+37 x^2+23 x+21$
• $y^2=12 x^6+16 x^5+12 x^4+42 x^3+2 x^2+37 x+25$
• $y^2=7 x^6+43 x^5+50 x^3+40 x^2+58 x+5$
• $y^2=21 x^6+5 x^5+15 x^4+12 x^3+21 x^2+41 x+10$

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 4.0.3187008.2.

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.as_ho	$2$	(not in LMFDB)