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

• Label: 2.59.as_hr
• 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:	$( 1 - 9 x + 59 x^{2} )^{2}$
	$1 - 18 x + 199 x^{2} - 1062 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.300760731311$, $\pm0.300760731311$
Angle rank:	$1$ (numerical)
Jacobians:	$34$
Cyclic group of points:	no
Non-cyclic primes:	$3, 17$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2601$	$12383361$	$42536587536$	$146966037932025$	$511115841057642921$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$42$	$3556$	$207108$	$12128548$	$714923022$	$42179862166$	$2488645516938$	$146830423510468$	$8662996043915292$	$511116756159521956$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 34 curves (of which all are hyperelliptic):

• $y^2=34 x^6+48 x^5+41 x^4+3 x^3+41 x^2+48 x+34$
• $y^2=29 x^6+35 x^5+26 x^4+27 x^3+5 x^2+4 x+35$
• $y^2=47 x^6+17 x^5+26 x^4+34 x^3+54 x^2+5 x+55$
• $y^2=43 x^6+20 x^5+45 x^4+46 x^3+45 x^2+20 x+43$
• $y^2=29 x^6+40 x^5+27 x^4+43 x^3+27 x^2+40 x+29$
• $y^2=43 x^6+51 x^5+2 x^4+42 x^3+55 x^2+37 x+55$
• $y^2=5 x^6+46 x^5+57 x^4+3 x^3+57 x^2+46 x+5$
• $y^2=30 x^6+34 x^5+44 x^4+53 x^3+17 x^2+4 x+55$
• $y^2=44 x^6+55 x^5+47 x^4+43 x^3+47 x^2+55 x+44$
• $y^2=50 x^6+42 x^5+8 x^4+51 x^2+36 x+40$
• $y^2=8 x^6+11 x^5+40 x^4+58 x^3+48 x^2+24 x+14$
• $y^2=32 x^6+50 x^5+48 x^4+31 x^3+48 x^2+50 x+32$
• $y^2=5 x^6+19 x^5+57 x^4+37 x^3+43 x^2+19 x+11$
• $y^2=10 x^6+50 x^5+15 x^4+10 x^3+29 x^2+23 x+38$
• $y^2=42 x^6+20 x^5+35 x^4+18 x^3+18 x^2+26 x+6$
• $y^2=51 x^6+26 x^5+12 x^4+10 x^3+12 x^2+26 x+51$
• $y^2=18 x^6+57 x^5+41 x^4+10 x^3+3 x^2+42 x+57$
• $y^2=38 x^6+42 x^5+19 x^4+18 x^3+19 x^2+42 x+38$
• $y^2=12 x^6+48 x^5+37 x^4+20 x^3+37 x^2+48 x+12$
• $y^2=54 x^6+56 x^5+13 x^4+18 x^3+38 x^2+40 x+33$
• and

• $y^2=58 x^6+34 x^5+21 x^4+24 x^3+19 x^2+38 x+37$
• $y^2=14 x^6+15 x^5+57 x^4+51 x^3+57 x^2+15 x+14$
• $y^2=34 x^6+31 x^5+51 x^4+11 x^3+51 x^2+31 x+34$
• $y^2=10 x^6+34 x^5+44 x^4+24 x^3+44 x^2+34 x+10$
• $y^2=33 x^6+35 x^5+38 x^4+33 x^3+38 x^2+35 x+33$
• $y^2=33 x^6+13 x^5+43 x^4+11 x^3+43 x^2+13 x+33$
• $y^2=4 x^6+2 x^5+30 x^4+51 x^3+30 x^2+2 x+4$
• $y^2=36 x^6+5 x^5+46 x^4+37 x^3+16 x^2+8 x+47$
• $y^2=38 x^6+52 x^4+41 x^3+47 x^2+23 x+28$
• $y^2=50 x^6+17 x^5+51 x^4+24 x^3+51 x^2+17 x+50$
• $y^2=34 x^6+43 x^5+18 x^4+16 x^3+18 x^2+43 x+34$
• $y^2=33 x^6+52 x^5+23 x^4+52 x^3+58 x^2+15 x+53$
• $y^2=44 x^6+50 x^5+42 x^4+42 x^3+42 x^2+50 x+44$
• $y^2=32 x^6+5 x^5+16 x^4+32 x^3+17 x^2+53 x+2$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.aj^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-155}) \)$)$

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_bl	$2$	(not in LMFDB)
2.59.s_hr	$2$	(not in LMFDB)
2.59.j_w	$3$	(not in LMFDB)