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

• Label: 2.59.y_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:	$( 1 + 10 x + 59 x^{2} )( 1 + 14 x + 59 x^{2} )$
	$1 + 24 x + 258 x^{2} + 1416 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.725626973200$, $\pm0.864937436951$
Angle rank:	$2$ (numerical)
Jacobians:	$32$
Isomorphism classes:	80

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5180$	$11914000$	$42077228060$	$146921541760000$	$511081769781251900$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$3422$	$204876$	$12124878$	$714875364$	$42180691502$	$2488651433436$	$146830441239838$	$8662995691476084$	$511116754938219902$

Jacobians and polarizations

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

• $y^2=19 x^6+10 x^5+31 x^4+7 x^3+31 x^2+10 x+19$
• $y^2=25 x^6+40 x^5+49 x^4+33 x^3+37 x^2+52 x+36$
• $y^2=30 x^6+x^5+16 x^4+32 x^3+51 x^2+16 x+34$
• $y^2=36 x^6+3 x^5+41 x^4+25 x^3+41 x^2+3 x+36$
• $y^2=29 x^6+16 x^5+27 x^4+39 x^3+27 x^2+16 x+29$
• $y^2=48 x^6+12 x^5+2 x^4+5 x^3+10 x^2+5 x+41$
• $y^2=39 x^6+38 x^5+23 x^4+27 x^3+40 x^2+44 x+14$
• $y^2=19 x^6+28 x^5+58 x^4+25 x^3+40 x^2+6 x+46$
• $y^2=57 x^6+6 x^5+32 x^4+6 x^3+32 x^2+6 x+57$
• $y^2=37 x^6+5 x^5+49 x^4+38 x^3+21 x^2+25 x+54$
• $y^2=29 x^6+13 x^5+55 x^4+28 x^3+38 x^2+8 x+12$
• $y^2=40 x^6+5 x^5+27 x^4+18 x^3+37 x+50$
• $y^2=8 x^6+34 x^5+23 x^4+45 x^3+51 x^2+43 x+35$
• $y^2=46 x^6+52 x^5+15 x^4+43 x^3+33 x^2+8 x+1$
• $y^2=17 x^6+9 x^5+33 x^4+23 x^3+19 x^2+21 x+27$
• $y^2=44 x^6+20 x^4+2 x^3+20 x^2+44$
• $y^2=49 x^6+57 x^5+40 x^4+21 x^3+37 x^2+37 x+35$
• $y^2=9 x^6+28 x^5+5 x^4+9 x^3+28 x^2+20 x+45$
• $y^2=49 x^6+44 x^5+52 x^4+6 x^3+14 x^2+5$
• $y^2=26 x^6+24 x^5+30 x^4+12 x^3+54 x^2+40 x+19$
• and

• $y^2=35 x^6+23 x^5+53 x^4+10 x^3+8 x^2+18 x+17$
• $y^2=15 x^6+11 x^5+14 x^4+21 x^3+8 x^2+6 x+20$
• $y^2=8 x^6+36 x^5+31 x^4+6 x^3+58 x^2+56 x+34$
• $y^2=27 x^6+12 x^5+48 x^4+47 x^3+36 x^2+8 x+35$
• $y^2=27 x^6+23 x^5+13 x^4+3 x^3+8 x^2+53 x+10$
• $y^2=19 x^6+38 x^5+17 x^4+5 x^3+17 x^2+38 x+19$
• $y^2=20 x^6+46 x^5+18 x^3+46 x+20$
• $y^2=19 x^6+19 x^5+58 x^4+51 x^3+23 x^2+21 x+48$
• $y^2=3 x^6+5 x^5+52 x^4+17 x^3+29 x^2+55 x+9$
• $y^2=57 x^6+36 x^5+20 x^4+45 x^3+20 x^2+36 x+57$
• $y^2=45 x^6+18 x^5+29 x^4+26 x^3+14 x^2+44 x+54$
• $y^2=58 x^6+30 x^5+15 x^4+19 x^3+26 x^2+39 x+30$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.k $\times$ 1.59.o 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.k : \(\Q(\sqrt{-34}) \).
• 1.59.o : \(\Q(\sqrt{-10}) \).

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.ay_jy	$2$	(not in LMFDB)
2.59.ae_aw	$2$	(not in LMFDB)
2.59.e_aw	$2$	(not in LMFDB)