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

• Label: 2.59.ay_jt
• 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:	$( 1 - 15 x + 59 x^{2} )( 1 - 9 x + 59 x^{2} )$
	$1 - 24 x + 253 x^{2} - 1416 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.0692665268586$, $\pm0.300760731311$
Angle rank:	$2$ (numerical)
Jacobians:	$28$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2295$	$11876625$	$42209897040$	$146843838545625$	$511098549913977975$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$3412$	$205524$	$12118468$	$714898836$	$42180090262$	$2488648350924$	$146830434658948$	$8662996001655756$	$511116755543649652$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):

• $y^2=25 x^6+35 x^5+11 x^4+29 x^3+33 x^2+20 x+26$
• $y^2=34 x^6+24 x^5+38 x^4+38 x^3+19 x^2+20 x+18$
• $y^2=58 x^6+53 x^5+43 x^4+49 x^3+30 x^2+25 x+18$
• $y^2=50 x^6+10 x^5+40 x^4+25 x^3+7 x^2+30 x+29$
• $y^2=52 x^6+55 x^5+16 x^4+21 x^3+34 x^2+34 x+42$
• $y^2=23 x^6+33 x^5+48 x^4+16 x^3+9 x^2+6$
• $y^2=32 x^6+14 x^5+x^4+41 x^3+7 x^2+35 x+24$
• $y^2=52 x^6+22 x^5+57 x^4+14 x^3+12 x^2+25 x+37$
• $y^2=58 x^6+20 x^5+44 x^4+33 x^3+11 x^2+54 x+58$
• $y^2=40 x^6+6 x^5+22 x^4+45 x^3+19 x^2+33 x+13$
• $y^2=42 x^6+49 x^5+43 x^4+57 x^3+23 x^2+x+23$
• $y^2=17 x^6+7 x^5+35 x^4+54 x^3+26 x^2+9 x+7$
• $y^2=41 x^6+12 x^5+5 x^4+43 x^3+51 x^2+7 x+36$
• $y^2=27 x^6+13 x^5+46 x^4+32 x^3+46 x^2+13 x+27$
• $y^2=17 x^6+39 x^5+36 x^4+18 x^3+12 x^2+24 x+5$
• $y^2=56 x^6+51 x^5+27 x^4+45 x^3+11 x^2+23 x+3$
• $y^2=6 x^6+53 x^5+17 x^4+54 x^3+7 x^2+49 x+18$
• $y^2=10 x^6+31 x^5+50 x^4+52 x^3+37 x^2+44 x+54$
• $y^2=18 x^6+19 x^5+21 x^4+5 x^3+10 x^2+45 x+43$
• $y^2=58 x^6+16 x^5+41 x^4+16 x^3+41 x^2+16 x+58$
• $y^2=17 x^6+x^5+43 x^4+23 x^3+33 x^2+22 x+3$
• $y^2=2 x^6+29 x^5+11 x^4+10 x^3+8 x^2+46 x+52$
• $y^2=6 x^6+21 x^5+52 x^4+35 x^3+37 x^2+20 x+42$
• $y^2=x^6+35 x^5+13 x^4+4 x^3+14 x^2+50 x+14$
• $y^2=38 x^6+33 x^5+6 x^4+41 x^3+14 x^2+42 x+2$
• $y^2=39 x^6+46 x^5+x^4+30 x^3+49 x^2+57 x+58$
• $y^2=44 x^6+18 x^5+x^4+3 x^3+37 x^2+38 x+33$
• $y^2=23 x^6+41 x^5+2 x^4+20 x^3+19 x^2+56 x+58$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.ap $\times$ 1.59.aj 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.ap : \(\Q(\sqrt{-11}) \).
• 1.59.aj : \(\Q(\sqrt{-155}) \).

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.ag_ar	$2$	(not in LMFDB)
2.59.g_ar	$2$	(not in LMFDB)
2.59.y_jt	$2$	(not in LMFDB)