Abelian variety isogeny class 2.113.ay_of over $\F_{113}$

• Label: 2.113.ay_of
• Base field: $\F_{113}$
• 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:	$\F_{113}$
Dimension:	$2$
L-polynomial:	$( 1 - 13 x + 113 x^{2} )( 1 - 11 x + 113 x^{2} )$
	$1 - 24 x + 369 x^{2} - 2712 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.290579079721$, $\pm0.326901256467$
Angle rank:	$2$ (numerical)
Jacobians:	$27$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10403$	$165147625$	$2088608811968$	$26590443047655625$	$339455425294634453123$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$90$	$12932$	$1447506$	$163084164$	$18424280490$	$2081946889694$	$235260498613482$	$26584441873827076$	$3004041942620989938$	$339456739051922969732$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 27 curves (of which all are hyperelliptic):

• $y^2=14 x^6+30 x^5+108 x^4+4 x^3+108 x^2+30 x+14$
• $y^2=40 x^6+91 x^5+45 x^4+57 x^3+45 x^2+91 x+40$
• $y^2=85 x^6+93 x^5+38 x^4+91 x^3+38 x^2+93 x+85$
• $y^2=19 x^6+49 x^5+90 x^4+60 x^3+90 x^2+49 x+19$
• $y^2=104 x^6+21 x^5+68 x^4+7 x^3+68 x^2+21 x+104$
• $y^2=38 x^6+14 x^5+38 x^4+75 x^3+38 x^2+14 x+38$
• $y^2=56 x^6+62 x^4+90 x^3+62 x^2+56$
• $y^2=43 x^6+72 x^5+112 x^4+26 x^3+112 x^2+72 x+43$
• $y^2=52 x^6+30 x^5+7 x^4+20 x^3+7 x^2+30 x+52$
• $y^2=19 x^6+82 x^5+64 x^4+5 x^3+64 x^2+82 x+19$
• $y^2=39 x^6+15 x^5+84 x^4+76 x^3+84 x^2+15 x+39$
• $y^2=107 x^6+66 x^5+65 x^4+80 x^3+65 x^2+66 x+107$
• $y^2=59 x^6+22 x^5+27 x^4+84 x^3+27 x^2+22 x+59$
• $y^2=46 x^6+15 x^5+51 x^4+50 x^3+51 x^2+15 x+46$
• $y^2=86 x^6+69 x^5+60 x^4+58 x^3+60 x^2+69 x+86$
• $y^2=63 x^6+24 x^5+86 x^4+53 x^3+86 x^2+24 x+63$
• $y^2=9 x^6+34 x^5+109 x^4+70 x^3+109 x^2+34 x+9$
• $y^2=56 x^6+x^5+61 x^4+39 x^3+61 x^2+x+56$
• $y^2=75 x^6+21 x^5+106 x^4+13 x^3+106 x^2+21 x+75$
• $y^2=48 x^6+94 x^5+38 x^4+103 x^3+38 x^2+94 x+48$
• $y^2=17 x^6+19 x^5+71 x^4+78 x^3+71 x^2+19 x+17$
• $y^2=72 x^6+42 x^5+90 x^4+106 x^3+90 x^2+42 x+72$
• $y^2=4 x^6+102 x^5+110 x^4+51 x^3+110 x^2+102 x+4$
• $y^2=91 x^6+102 x^5+56 x^4+28 x^3+56 x^2+102 x+91$
• $y^2=45 x^6+75 x^5+79 x^4+109 x^3+79 x^2+75 x+45$
• $y^2=39 x^6+3 x^5+10 x^4+69 x^3+10 x^2+3 x+39$
• $y^2=37 x^6+10 x^5+86 x^4+53 x^3+86 x^2+10 x+37$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{113}$.

Endomorphism algebra over $\F_{113}$

The isogeny class factors as 1.113.an $\times$ 1.113.al and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.113.an : \(\Q(\sqrt{-283}) \).
• 1.113.al : \(\Q(\sqrt{-331}) \).

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
2.113.ac_df	$2$	(not in LMFDB)
2.113.c_df	$2$	(not in LMFDB)
2.113.y_of	$2$	(not in LMFDB)