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

• Label: 2.59.y_kc
• 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:	$( 1 + 12 x + 59 x^{2} )^{2}$
	$1 + 24 x + 262 x^{2} + 1416 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.785358177425$, $\pm0.785358177425$
Angle rank:	$1$ (numerical)
Jacobians:	$33$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5184$	$11943936$	$42018440256$	$146982840827904$	$511042308084661824$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$3430$	$204588$	$12129934$	$714820164$	$42181041526$	$2488651534236$	$146830407046174$	$8662996182437172$	$511116750738185350$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 33 curves (of which all are hyperelliptic):

• $y^2=22 x^6+11 x^5+24 x^4+43 x^3+13 x^2+40 x+4$
• $y^2=49 x^6+40 x^5+5 x^4+58 x^3+7 x^2+43 x+41$
• $y^2=31 x^6+25 x^5+23 x^4+24 x^3+23 x^2+25 x+31$
• $y^2=46 x^6+30 x^5+43 x^4+7 x^3+13 x^2+47 x+12$
• $y^2=56 x^6+24 x^5+42 x^4+45 x^3+42 x^2+24 x+56$
• $y^2=38 x^6+23 x^5+58 x^4+47 x^3+58 x^2+23 x+38$
• $y^2=21 x^6+6 x^5+27 x^4+44 x^3+21 x^2+8 x+48$
• $y^2=57 x^6+13 x^5+57 x^4+38 x^3+19 x^2+8 x+48$
• $y^2=49 x^6+34 x^5+14 x^4+19 x^3+14 x^2+34 x+49$
• $y^2=30 x^6+17 x^5+16 x^4+27 x^3+16 x^2+17 x+30$
• $y^2=14 x^6+56 x^4+56 x^2+14$
• $y^2=26 x^6+15 x^5+27 x^4+27 x^3+29 x^2+36 x+29$
• $y^2=25 x^6+29 x^5+58 x^4+3 x^3+32 x^2+19 x+15$
• $y^2=33 x^6+23 x^5+13 x^4+30 x^3+13 x^2+23 x+33$
• $y^2=49 x^6+29 x^5+30 x^4+46 x^3+30 x^2+29 x+49$
• $y^2=57 x^6+3 x^5+13 x^4+27 x^3+37 x^2+25 x+4$
• $y^2=35 x^6+53 x^5+47 x^4+22 x^3+47 x^2+53 x+35$
• $y^2=19 x^6+31 x^4+31 x^2+19$
• $y^2=29 x^6+7 x^5+55 x^3+16 x+49$
• $y^2=8 x^5+14 x^4+55 x^3+38 x^2+18 x$
• and

• $y^2=18 x^6+44 x^4+44 x^2+18$
• $y^2=3 x^6+26 x^5+36 x^4+55 x^3+25 x^2+17 x+25$
• $y^2=41 x^6+47 x^5+3 x^4+21 x^3+3 x^2+47 x+41$
• $y^2=12 x^6+7 x^4+7 x^2+12$
• $y^2=23 x^6+18 x^5+21 x^4+2 x^3+25 x^2+52 x+56$
• $y^2=30 x^6+22 x^5+43 x^4+42 x^3+43 x^2+22 x+30$
• $y^2=10 x^5+14 x^4+4 x^3+14 x^2+10 x$
• $y^2=55 x^6+28 x^4+28 x^2+55$
• $y^2=29 x^6+17 x^5+26 x^4+50 x^3+29 x^2+16 x+51$
• $y^2=20 x^6+37 x^5+20 x^4+8 x^3+x^2+38 x+9$
• $y^2=34 x^6+56 x^5+27 x^4+56 x^3+25 x^2+54 x+43$
• $y^2=7 x^6+32 x^4+32 x^2+7$
• $y^2=46 x^6+49 x^5+13 x^4+11 x^3+13 x^2+49 x+46$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.m^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-23}) \)$)$

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.ay_kc	$2$	(not in LMFDB)
2.59.a_aba	$2$	(not in LMFDB)
2.59.am_dh	$3$	(not in LMFDB)