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

• Label: 2.59.aw_iy
• 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:	$1 - 22 x + 232 x^{2} - 1298 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.151912498280$, $\pm0.316979219896$
Angle rank:	$2$ (numerical)
Number field:	4.0.6488384.1
Galois group:	$D_{4}$
Jacobians:	$40$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2394$	$12051396$	$42338974482$	$146914663983696$	$511134511471541874$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$38$	$3462$	$206150$	$12124310$	$714949138$	$42180537222$	$2488652139298$	$146830459226014$	$8662996060679030$	$511116754445989302$

Jacobians and polarizations

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

• $y^2=30 x^6+20 x^5+8 x^3+49 x^2+57 x+19$
• $y^2=31 x^6+6 x^5+51 x^4+33 x^3+24 x^2+44 x+11$
• $y^2=18 x^6+15 x^5+35 x^4+5 x^3+29 x^2+49 x+24$
• $y^2=44 x^6+29 x^5+48 x^4+23 x^3+45 x^2+54 x+28$
• $y^2=19 x^6+11 x^5+40 x^4+20 x^3+13 x^2+36 x+8$
• $y^2=52 x^6+52 x^5+17 x^4+33 x^3+57 x^2+10 x+47$
• $y^2=40 x^6+33 x^5+41 x^4+48 x^3+x^2+3 x+54$
• $y^2=6 x^6+18 x^5+58 x^4+45 x^3+9 x^2+13 x+46$
• $y^2=3 x^6+51 x^5+19 x^4+50 x^3+57 x^2+3 x+27$
• $y^2=52 x^6+12 x^5+58 x^4+13 x^3+55 x+14$
• $y^2=37 x^6+32 x^5+27 x^4+24 x^3+17 x^2+45 x+25$
• $y^2=23 x^6+24 x^5+41 x^4+50 x^3+9 x^2+25 x+23$
• $y^2=43 x^6+26 x^5+16 x^4+33 x^3+13 x^2+58 x+20$
• $y^2=58 x^6+9 x^5+52 x^4+37 x^3+52 x^2+33 x+54$
• $y^2=44 x^6+26 x^5+57 x^4+39 x^3+40 x^2+34 x+29$
• $y^2=44 x^6+13 x^5+49 x^4+32 x^3+23 x^2+40$
• $y^2=54 x^6+29 x^5+15 x^4+23 x^3+15 x^2+50 x+47$
• $y^2=30 x^6+58 x^5+22 x^4+48 x^3+11 x^2+16$
• $y^2=33 x^6+42 x^5+24 x^4+43 x^3+7 x^2+10 x+23$
• $y^2=34 x^6+15 x^5+47 x^4+21 x^3+49 x^2+58 x+43$
• and

• $y^2=38 x^6+3 x^5+13 x^4+51 x^3+42 x^2+3 x+48$
• $y^2=23 x^6+55 x^5+22 x^4+30 x^3+31 x^2+34 x+32$
• $y^2=3 x^6+49 x^5+6 x^4+27 x^3+28 x^2+18 x+36$
• $y^2=38 x^6+55 x^5+x^4+31 x^3+45 x^2+26 x+11$
• $y^2=5 x^6+31 x^5+56 x^4+54 x^3+35 x^2+17 x+44$
• $y^2=43 x^6+25 x^5+50 x^4+49 x^3+30 x^2+21 x+9$
• $y^2=55 x^6+58 x^5+10 x^4+18 x^3+30 x^2+47 x+55$
• $y^2=26 x^6+24 x^5+56 x^4+56 x^3+2 x^2+40 x+26$
• $y^2=44 x^6+5 x^5+41 x^4+57 x^3+17 x^2+50 x+33$
• $y^2=43 x^6+34 x^5+27 x^4+10 x^3+32 x^2+34 x+6$
• $y^2=11 x^6+41 x^5+15 x^4+56 x^3+50 x^2+33 x+24$
• $y^2=24 x^6+18 x^5+50 x^4+3 x^3+7 x^2+35 x+13$
• $y^2=48 x^6+13 x^5+31 x^4+22 x^3+39 x^2+24 x+45$
• $y^2=30 x^6+38 x^5+17 x^4+11 x^3+57 x^2+33 x+56$
• $y^2=55 x^6+8 x^5+41 x^4+25 x^3+27 x^2+27 x+29$
• $y^2=16 x^6+7 x^5+2 x^4+38 x^3+3 x^2+42 x+18$
• $y^2=34 x^6+28 x^5+27 x^4+21 x^3+58 x^2+11 x+52$
• $y^2=58 x^6+56 x^5+54 x^4+56 x^3+36 x^2+25 x+58$
• $y^2=14 x^6+26 x^5+29 x^4+41 x^3+3 x^2+28 x+12$
• $y^2=50 x^6+47 x^5+15 x^4+46 x^3+54 x^2+52 x+6$

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 4.0.6488384.1.

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.w_iy	$2$	(not in LMFDB)