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

• Label: 2.59.o_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:	$\pm0.650597135156$, $\pm0.650597135156$
Angle rank:	$1$ (numerical)
Jacobians:	$17$
Cyclic group of points:	no
Non-cyclic primes:	$67$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4489$	$12609601$	$41813706256$	$146883807310969$	$511170313972527049$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$74$	$3620$	$203588$	$12121764$	$714999214$	$42179749526$	$2488652553706$	$146830476384964$	$8662995484126172$	$511116753354284900$

Jacobians and polarizations

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

• $y^2=3 x^6+32 x^5+21 x^4+51 x^3+2 x^2+27 x+52$
• $y^2=29 x^6+52 x^5+44 x^4+26 x^3+44 x^2+52 x+29$
• $y^2=2 x^6+58 x^5+31 x^4+9 x^3+5 x^2+50 x+28$
• $y^2=48 x^6+29 x^5+46 x^4+43 x^3+22 x^2+47 x+10$
• $y^2=21 x^6+2 x^5+27 x^4+23 x^3+54 x^2+20 x+54$
• $y^2=17 x^6+11 x^5+40 x^4+29 x^3+40 x^2+11 x+17$
• $y^2=20 x^6+15 x^5+43 x^4+28 x^3+57 x^2+49$
• $y^2=41 x^6+19 x^5+25 x^4+56 x^3+55 x^2+18 x+45$
• $y^2=x^6+10 x^5+35 x^4+37 x^3+2 x^2+34 x+30$
• $y^2=48 x^6+22 x^5+42 x^3+22 x+48$
• $y^2=49 x^6+58 x^5+7 x^4+24 x^3+52 x^2+54 x+18$
• $y^2=8 x^6+6 x^5+10 x^4+39 x^3+5 x^2+19 x+21$
• $y^2=10 x^6+25 x^5+46 x^4+32 x^3+46 x^2+25 x+10$
• $y^2=56 x^6+3 x^5+9 x^4+51 x^3+13 x^2+24 x+51$
• $y^2=4 x^6+11 x^5+55 x^4+36 x^3+55 x^2+11 x+4$
• $y^2=17 x^6+32 x^5+30 x^4+26 x^3+15 x^2+18 x+56$
• $y^2=7 x^6+45 x^5+34 x^4+7 x^3+48 x^2+41 x+43$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.h^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.ao_gl	$2$	(not in LMFDB)
2.59.a_cr	$2$	(not in LMFDB)
2.59.ah_ak	$3$	(not in LMFDB)