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

• Label: 2.59.az_kk
• 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:	$1 - 25 x + 270 x^{2} - 1475 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.103225557456$, $\pm0.262203249885$
Angle rank:	$2$ (numerical)
Number field:	4.0.219929.1
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	12

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2252$	$11827504$	$42221658032$	$146891023587776$	$511140079216801252$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$35$	$3397$	$205580$	$12122361$	$714956925$	$42180590422$	$2488650772895$	$146830435195953$	$8662995920848340$	$511116755257995477$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

• $y^2=38 x^6+24 x^5+22 x^4+25 x^3+45 x^2+26 x+53$
• $y^2=42 x^6+35 x^5+15 x^4+22 x^3+36 x^2+29 x+37$
• $y^2=10 x^6+28 x^5+41 x^4+51 x^3+45 x^2+14 x+24$
• $y^2=6 x^6+16 x^5+17 x^4+5 x^3+5 x^2+17 x+43$
• $y^2=30 x^6+2 x^5+52 x^4+11 x^3+51 x^2+57 x+4$
• $y^2=14 x^6+42 x^5+5 x^4+23 x^3+46 x^2+53 x+33$
• $y^2=27 x^5+14 x^4+31 x^3+30 x^2+34 x+52$
• $y^2=44 x^6+49 x^5+46 x^4+8 x^3+41 x^2+21 x+6$
• $y^2=22 x^6+2 x^5+54 x^4+41 x^3+38 x^2+55 x+42$
• $y^2=47 x^6+28 x^5+55 x^4+8 x^3+4 x^2+35 x+50$
• $y^2=24 x^6+56 x^5+20 x^4+31 x^3+7 x^2+20 x+33$
• $y^2=44 x^6+34 x^5+47 x^4+43 x^3+45 x^2+15 x+38$

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 4.0.219929.1.

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_kk	$2$	(not in LMFDB)