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

• Label: 2.59.ah_ez
• 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:	$1 - 7 x + 129 x^{2} - 413 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.402810282391$, $\pm0.450445236886$
Angle rank:	$2$ (numerical)
Number field:	4.0.1236125.2
Galois group:	$D_{4}$
Jacobians:	$16$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3191$	$12862921$	$42412701041$	$146733063846845$	$511051718643034096$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$53$	$3691$	$206507$	$12109323$	$714833328$	$42180671791$	$2488656944057$	$146830448410083$	$8662995563782763$	$511116751896361206$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):

• $y^2=x^6+8 x^5+47 x^4+33 x^3+27 x^2+36 x+57$
• $y^2=34 x^6+22 x^5+40 x^4+23 x^3+22 x^2+10 x+55$
• $y^2=34 x^6+31 x^5+18 x^4+15 x^2+10 x+33$
• $y^2=31 x^6+41 x^5+48 x^4+10 x^3+30 x^2+17 x+57$
• $y^2=48 x^6+44 x^5+10 x^4+45 x^3+53 x^2+7 x+49$
• $y^2=22 x^6+24 x^5+52 x^4+19 x^3+20 x+36$
• $y^2=52 x^6+6 x^5+6 x^4+31 x^3+6 x^2+17 x+14$
• $y^2=18 x^6+58 x^5+x^4+40 x^3+18 x^2+48 x+47$
• $y^2=15 x^6+20 x^5+25 x^4+38 x^3+11 x^2+21 x+6$
• $y^2=56 x^6+58 x^5+9 x^4+37 x^3+34 x^2+2 x+32$
• $y^2=3 x^6+25 x^5+6 x^4+40 x^3+58 x^2+x+19$
• $y^2=44 x^6+41 x^5+45 x^4+24 x^3+8 x^2+15 x+44$
• $y^2=34 x^6+34 x^5+39 x^4+6 x^3+8 x^2+17 x+48$
• $y^2=42 x^6+13 x^5+54 x^4+51 x^3+29 x^2+53 x+35$
• $y^2=9 x^6+14 x^5+30 x^4+50 x^3+2 x^2+47 x+51$
• $y^2=52 x^6+38 x^5+31 x^4+22 x^3+18 x^2+43 x+13$

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 4.0.1236125.2.

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.h_ez	$2$	(not in LMFDB)