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

• Label: 2.59.ao_gl
• Base field: $\F_{59}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$( 1 - 7 x + 59 x^{2} )^{2}$
	$1 - 14 x + 167 x^{2} - 826 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.349402864844$, $\pm0.349402864844$
Angle rank:	$1$ (numerical)
Jacobians:	$17$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2809$	$12609601$	$42549788176$	$146883807310969$	$511063198294506649$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$46$	$3620$	$207172$	$12121764$	$714849386$	$42179749526$	$2488650415934$	$146830476384964$	$8662996153183708$	$511116753354284900$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 17 curves (of which all are hyperelliptic):

• $y^2=31 x^6+16 x^5+40 x^4+55 x^3+x^2+43 x+26$
• $y^2=44 x^6+26 x^5+22 x^4+13 x^3+22 x^2+26 x+44$
• $y^2=x^6+29 x^5+45 x^4+34 x^3+32 x^2+25 x+14$
• $y^2=37 x^6+58 x^5+33 x^4+27 x^3+44 x^2+35 x+20$
• $y^2=40 x^6+x^5+43 x^4+41 x^3+27 x^2+10 x+27$
• $y^2=38 x^6+35 x^5+20 x^4+44 x^3+20 x^2+35 x+38$
• $y^2=10 x^6+37 x^5+51 x^4+14 x^3+58 x^2+54$
• $y^2=23 x^6+38 x^5+50 x^4+53 x^3+51 x^2+36 x+31$
• $y^2=30 x^6+5 x^5+47 x^4+48 x^3+x^2+17 x+15$
• $y^2=37 x^6+44 x^5+25 x^3+44 x+37$
• $y^2=54 x^6+29 x^5+33 x^4+12 x^3+26 x^2+27 x+9$
• $y^2=16 x^6+12 x^5+20 x^4+19 x^3+10 x^2+38 x+42$
• $y^2=20 x^6+50 x^5+33 x^4+5 x^3+33 x^2+50 x+20$
• $y^2=53 x^6+6 x^5+18 x^4+43 x^3+26 x^2+48 x+43$
• $y^2=2 x^6+35 x^5+57 x^4+18 x^3+57 x^2+35 x+2$
• $y^2=34 x^6+5 x^5+x^4+52 x^3+30 x^2+36 x+53$
• $y^2=33 x^6+52 x^5+17 x^4+33 x^3+24 x^2+50 x+51$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{59}$.

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.ah^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-187}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.59.a_cr	$2$	(not in LMFDB)
2.59.o_gl	$2$	(not in LMFDB)
2.59.h_ak	$3$	(not in LMFDB)