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

• Label: 2.59.au_ik
• 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:	$( 1 - 10 x + 59 x^{2} )^{2}$
	$1 - 20 x + 218 x^{2} - 1180 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.274373026800$, $\pm0.274373026800$
Angle rank:	$1$ (numerical)
Jacobians:	$26$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2500$	$12250000$	$42497822500$	$146991376000000$	$511146710497562500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$40$	$3518$	$206920$	$12130638$	$714966200$	$42180169358$	$2488645369880$	$146830397947678$	$8662995782869960$	$511116755282533598$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 26 curves (of which all are hyperelliptic):

• $y^2=5 x^6+24 x^5+51 x^4+26 x^3+57 x^2+31 x+1$
• $y^2=3 x^6+28 x^5+22 x^4+28 x^3+29 x^2+35 x+15$
• $y^2=51 x^6+43 x^5+17 x^4+56 x^3+27 x^2+56 x+41$
• $y^2=11 x^6+18 x^5+57 x^4+35 x^3+23 x^2+x+37$
• $y^2=44 x^6+5 x^5+9 x^4+50 x^3+42 x^2+6 x+35$
• $y^2=26 x^6+57 x^5+7 x^4+9 x^3+53 x^2+16 x+40$
• $y^2=42 x^6+58 x^5+58 x^4+21 x^3+38 x^2+5 x+32$
• $y^2=37 x^6+3 x^5+9 x^4+55 x^3+9 x^2+3 x+37$
• $y^2=35 x^6+4 x^5+53 x^4+42 x^3+28 x^2+15 x+21$
• $y^2=41 x^5+55 x^4+51 x^3+56 x^2+37 x+45$
• $y^2=53 x^5+54 x^4+36 x^3+54 x^2+53 x$
• $y^2=34 x^6+15 x^4+15 x^2+34$
• $y^2=9 x^6+42 x^5+33 x^4+43 x^3+33 x^2+42 x+9$
• $y^2=32 x^6+7 x^5+4 x^4+55 x^3+25 x^2+19 x+54$
• $y^2=32 x^6+47 x^5+30 x^4+33 x^3+37 x^2+56 x+14$
• $y^2=31 x^6+54 x^5+50 x^4+18 x^3+50 x^2+54 x+31$
• $y^2=36 x^6+3 x^5+5 x^4+45 x^3+18 x^2+47 x+54$
• $y^2=28 x^6+50 x^5+3 x^4+40 x^3+15 x^2+15 x+52$
• $y^2=49 x^6+48 x^5+9 x^4+52 x^3+24 x^2+34 x+19$
• $y^2=39 x^6+45 x^5+20 x^4+55 x^3+20 x^2+45 x+39$
• $y^2=14 x^6+53 x^5+51 x^4+43 x^3+43 x^2+49 x+50$
• $y^2=43 x^5+47 x^4+8 x^3+31 x^2+49 x+2$
• $y^2=36 x^6+18 x^5+x^4+31 x^3+x^2+18 x+36$
• $y^2=51 x^6+24 x^5+30 x^4+2 x^3+33 x^2+55 x+29$
• $y^2=11 x^6+33 x^5+39 x^4+12 x^3+39 x^2+33 x+11$
• $y^2=53 x^6+46 x^4+46 x^2+53$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.ak^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-34}) \)$)$

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_s	$2$	(not in LMFDB)
2.59.u_ik	$2$	(not in LMFDB)
2.59.k_bp	$3$	(not in LMFDB)