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

• Label: 2.59.ay_jy
• 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:	$( 1 - 14 x + 59 x^{2} )( 1 - 10 x + 59 x^{2} )$
	$1 - 24 x + 258 x^{2} - 1416 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.135062563049$, $\pm0.274373026800$
Angle rank:	$2$ (numerical)
Jacobians:	$32$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2300$	$11914000$	$42284251100$	$146921541760000$	$511151740852341500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$3422$	$205884$	$12124878$	$714973236$	$42180691502$	$2488651536204$	$146830441239838$	$8662995945833796$	$511116754938219902$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 32 curves (of which all are hyperelliptic):

• $y^2=38 x^6+20 x^5+3 x^4+14 x^3+3 x^2+20 x+38$
• $y^2=42 x^6+20 x^5+54 x^4+46 x^3+48 x^2+26 x+18$
• $y^2=15 x^6+30 x^5+8 x^4+16 x^3+55 x^2+8 x+17$
• $y^2=18 x^6+31 x^5+50 x^4+42 x^3+50 x^2+31 x+18$
• $y^2=44 x^6+8 x^5+43 x^4+49 x^3+43 x^2+8 x+44$
• $y^2=37 x^6+24 x^5+4 x^4+10 x^3+20 x^2+10 x+23$
• $y^2=19 x^6+17 x^5+46 x^4+54 x^3+21 x^2+29 x+28$
• $y^2=39 x^6+14 x^5+29 x^4+42 x^3+20 x^2+3 x+23$
• $y^2=55 x^6+12 x^5+5 x^4+12 x^3+5 x^2+12 x+55$
• $y^2=48 x^6+32 x^5+54 x^4+19 x^3+40 x^2+42 x+27$
• $y^2=58 x^6+26 x^5+51 x^4+56 x^3+17 x^2+16 x+24$
• $y^2=21 x^6+10 x^5+54 x^4+36 x^3+15 x+41$
• $y^2=16 x^6+9 x^5+46 x^4+31 x^3+43 x^2+27 x+11$
• $y^2=33 x^6+45 x^5+30 x^4+27 x^3+7 x^2+16 x+2$
• $y^2=34 x^6+18 x^5+7 x^4+46 x^3+38 x^2+42 x+54$
• $y^2=29 x^6+40 x^4+4 x^3+40 x^2+29$
• $y^2=39 x^6+55 x^5+21 x^4+42 x^3+15 x^2+15 x+11$
• $y^2=34 x^6+14 x^5+32 x^4+34 x^3+14 x^2+10 x+52$
• $y^2=39 x^6+29 x^5+45 x^4+12 x^3+28 x^2+10$
• $y^2=52 x^6+48 x^5+x^4+24 x^3+49 x^2+21 x+38$
• and

• $y^2=47 x^6+41 x^5+56 x^4+5 x^3+4 x^2+9 x+38$
• $y^2=30 x^6+22 x^5+28 x^4+42 x^3+16 x^2+12 x+40$
• $y^2=16 x^6+13 x^5+3 x^4+12 x^3+57 x^2+53 x+9$
• $y^2=43 x^6+6 x^5+24 x^4+53 x^3+18 x^2+4 x+47$
• $y^2=54 x^6+46 x^5+26 x^4+6 x^3+16 x^2+47 x+20$
• $y^2=38 x^6+17 x^5+34 x^4+10 x^3+34 x^2+17 x+38$
• $y^2=10 x^6+23 x^5+9 x^3+23 x+10$
• $y^2=39 x^6+39 x^5+29 x^4+55 x^3+41 x^2+40 x+24$
• $y^2=31 x^6+32 x^5+26 x^4+38 x^3+44 x^2+57 x+34$
• $y^2=55 x^6+13 x^5+40 x^4+31 x^3+40 x^2+13 x+55$
• $y^2=52 x^6+9 x^5+44 x^4+13 x^3+7 x^2+22 x+27$
• $y^2=57 x^6+x^5+30 x^4+38 x^3+52 x^2+19 x+1$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.ao $\times$ 1.59.ak and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.59.ao : \(\Q(\sqrt{-10}) \).
• 1.59.ak : \(\Q(\sqrt{-34}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.59.ae_aw	$2$	(not in LMFDB)
2.59.e_aw	$2$	(not in LMFDB)
2.59.y_jy	$2$	(not in LMFDB)