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

• Label: 2.59.az_kl
• 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:	$1 - 25 x + 271 x^{2} - 1475 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.118910193748$, $\pm0.254815024423$
Angle rank:	$2$ (numerical)
Number field:	4.0.645749.1
Galois group:	$D_{4}$
Jacobians:	$11$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2253$	$11835009$	$42237159975$	$146908215252189$	$511152858953081328$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$35$	$3399$	$205655$	$12123779$	$714974800$	$42180751683$	$2488651758145$	$146830437498259$	$8662995887752265$	$511116754700971254$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 11 curves (of which all are hyperelliptic):

• $y^2=34 x^6+30 x^5+39 x^4+39 x^3+33 x^2+17 x+47$
• $y^2=6 x^6+51 x^5+42 x^4+49 x^3+24 x^2+25 x+50$
• $y^2=x^5+34 x^4+21 x^3+11 x^2+5 x+34$
• $y^2=7 x^6+17 x^5+57 x^4+26 x^3+33 x^2+55 x+43$
• $y^2=29 x^6+22 x^5+41 x^4+11 x^3+14 x^2+44 x+40$
• $y^2=21 x^6+12 x^5+11 x^4+49 x^3+40 x^2+57 x+43$
• $y^2=31 x^5+58 x^4+31 x^3+44 x^2+31 x+31$
• $y^2=37 x^6+57 x^5+26 x^4+13 x^3+17 x^2+45 x+37$
• $y^2=33 x^6+19 x^5+35 x^4+20 x^3+37 x^2+58 x+40$
• $y^2=54 x^6+12 x^5+36 x^4+18 x^3+32 x^2+2 x+38$
• $y^2=2 x^6+23 x^5+58 x^4+49 x^3+4 x^2+30 x+21$

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 4.0.645749.1.

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.z_kl	$2$	(not in LMFDB)