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

• Label: 2.59.aq_ha
• 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:	$( 1 - 8 x + 59 x^{2} )^{2}$
	$1 - 16 x + 182 x^{2} - 944 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.325650265238$, $\pm0.325650265238$
Angle rank:	$1$ (numerical)
Jacobians:	$30$
Cyclic group of points:	no
Non-cyclic primes:	$2, 13$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2704$	$12503296$	$42553088656$	$146928531902464$	$511086774104702224$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$44$	$3590$	$207188$	$12125454$	$714882364$	$42179720726$	$2488647455716$	$146830453333534$	$8662996182205772$	$511116755281024550$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):

• $y^2=55 x^6+7 x^5+3 x^4+29 x^3+20$
• $y^2=43 x^6+29 x^5+x^4+4 x^3+49 x^2+51 x+46$
• $y^2=16 x^6+12 x^5+38 x^4+5 x^3+13 x^2+20 x+25$
• $y^2=20 x^6+x^5+55 x^4+50 x^3+55 x^2+x+20$
• $y^2=x^6+37 x^5+26 x^4+7 x^3+26 x^2+37 x+1$
• $y^2=50 x^6+56 x^5+20 x^4+8 x^3+20 x^2+56 x+50$
• $y^2=21 x^6+36 x^5+33 x^4+23 x^3+48 x^2+28 x+24$
• $y^2=27 x^6+51 x^5+38 x^4+56 x^3+50 x^2+41 x+20$
• $y^2=57 x^6+18 x^4+18 x^2+57$
• $y^2=42 x^6+18 x^5+58 x^4+5 x^3+21 x^2+3 x+28$
• $y^2=11 x^6+12 x^4+12 x^2+11$
• $y^2=3 x^6+36 x^5+16 x^4+49 x^3+45 x^2+10 x+2$
• $y^2=58 x^6+23 x^5+37 x^4+33 x^3+42 x^2+29 x+9$
• $y^2=58 x^6+20 x^5+43 x^4+44 x^3+3 x^2+57 x+10$
• $y^2=12 x^6+56 x^5+48 x^4+24 x^3+17 x^2+24 x+22$
• $y^2=6 x^6+46 x^5+24 x^4+31 x^3+13 x^2+x+44$
• $y^2=38 x^6+23 x^5+8 x^4+35 x^3+30 x^2+27 x+16$
• $y^2=13 x^6+46 x^5+30 x^4+29 x^3+30 x^2+46 x+13$
• $y^2=43 x^6+7 x^5+58 x^4+34 x^3+58 x^2+7 x+43$
• $y^2=15 x^6+30 x^5+35 x^4+8 x^3+57 x^2+42 x+46$
• $y^2=7 x^6+56 x^5+4 x^4+31 x^3+34 x^2+26 x+25$
• $y^2=14 x^6+47 x^5+6 x^4+12 x^3+16 x^2+32 x+2$
• $y^2=35 x^6+10 x^4+10 x^2+35$
• $y^2=28 x^6+47 x^5+3 x^4+18 x^3+18 x^2+44 x+6$
• $y^2=7 x^6+58 x^5+22 x^4+55 x^3+14 x^2+2 x+30$
• $y^2=42 x^6+32 x^5+54 x^3+33 x^2+17 x+7$
• $y^2=51 x^5+21 x^4+36 x^3+53 x^2+5 x+18$
• $y^2=8 x^6+5 x^5+5 x^4+57 x^3+48 x^2+57 x+22$
• $y^2=9 x^6+8 x^4+54 x^3+58 x^2+33 x+54$
• $y^2=20 x^6+54 x^5+55 x^4+39 x^3+41 x^2+18 x+43$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.ai^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$

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_cc	$2$	(not in LMFDB)
2.59.q_ha	$2$	(not in LMFDB)
2.59.i_f	$3$	(not in LMFDB)