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

• Label: 2.113.ay_nk
• Base field: $\F_{113}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{113}$
Dimension:	$2$
L-polynomial:	$1 - 24 x + 348 x^{2} - 2712 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.212636630458$, $\pm0.388281055844$
Angle rank:	$2$ (numerical)
Number field:	4.0.535296256.1
Galois group:	$D_{4}$
Jacobians:	$60$
Isomorphism classes:	60

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10382$	$164596228$	$2086423955966$	$26587463752253584$	$339456725250323649182$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$90$	$12890$	$1445994$	$163065894$	$18424351050$	$2081952149690$	$235260568559610$	$26584442044582654$	$3004041934703941722$	$339456738923859899450$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 60 curves (of which all are hyperelliptic):

• $y^2=78 x^6+79 x^5+49 x^4+65 x^3+108 x^2+26 x+94$
• $y^2=5 x^6+52 x^5+54 x^4+39 x^3+108 x^2+16 x+13$
• $y^2=3 x^6+4 x^5+12 x^4+24 x^3+6 x^2+71 x+104$
• $y^2=75 x^6+40 x^5+68 x^4+98 x^3+47 x^2+104 x+47$
• $y^2=47 x^6+54 x^4+39 x^3+75 x^2+x+42$
• $y^2=76 x^6+99 x^5+50 x^4+76 x^3+6 x^2+12 x+75$
• $y^2=45 x^6+30 x^5+110 x^4+x^3+58 x^2+71 x+101$
• $y^2=92 x^6+94 x^5+16 x^4+11 x^3+4 x^2+111 x+67$
• $y^2=8 x^6+48 x^5+13 x^4+48 x^3+82 x^2+40 x+65$
• $y^2=25 x^6+67 x^5+68 x^4+40 x^3+5 x^2+45 x+39$
• $y^2=79 x^6+98 x^5+53 x^4+112 x^3+80 x^2+45 x+8$
• $y^2=100 x^6+99 x^5+74 x^4+27 x^3+36 x^2+45 x+76$
• $y^2=22 x^6+77 x^5+12 x^4+42 x^3+105 x^2+39 x+2$
• $y^2=99 x^6+65 x^5+74 x^4+109 x^3+11 x^2+68 x+111$
• $y^2=18 x^6+100 x^5+93 x^4+50 x^3+72 x^2+76 x+26$
• $y^2=59 x^6+99 x^5+4 x^4+43 x^3+28 x^2+88 x+56$
• $y^2=6 x^6+7 x^5+90 x^4+84 x^3+84 x^2+27 x+37$
• $y^2=58 x^6+22 x^5+103 x^4+40 x^3+16 x^2+17 x+48$
• $y^2=33 x^6+78 x^5+96 x^4+76 x^3+65 x^2+17 x+57$
• $y^2=107 x^6+45 x^5+67 x^4+50 x^3+7 x^2+83 x+60$
• and

• $y^2=46 x^6+92 x^5+68 x^4+51 x^3+82 x^2+59 x+109$
• $y^2=16 x^6+30 x^5+8 x^4+14 x^3+108 x^2+44 x+95$
• $y^2=109 x^6+43 x^5+89 x^4+103 x^3+95 x^2+23 x+68$
• $y^2=10 x^6+83 x^5+107 x^4+112 x^3+102 x^2+69 x+112$
• $y^2=79 x^6+5 x^5+70 x^4+112 x^3+56 x^2+59 x+5$
• $y^2=86 x^6+58 x^5+86 x^4+101 x^3+85 x^2+50 x+81$
• $y^2=x^6+41 x^5+67 x^4+29 x^3+99 x^2+108 x+84$
• $y^2=28 x^6+74 x^5+71 x^4+2 x^3+76 x^2+63 x+27$
• $y^2=96 x^6+101 x^5+71 x^4+47 x^3+42 x^2+14 x+60$
• $y^2=74 x^6+31 x^5+93 x^4+52 x^3+7 x^2+73 x+73$
• $y^2=70 x^6+56 x^5+86 x^4+76 x^3+17 x^2+91 x+101$
• $y^2=53 x^6+94 x^5+39 x^4+10 x^3+33 x^2+39 x+55$
• $y^2=10 x^6+46 x^5+13 x^4+57 x^3+105 x^2+85 x+40$
• $y^2=101 x^6+67 x^5+68 x^4+48 x^3+76 x^2+85 x+75$
• $y^2=74 x^6+101 x^5+57 x^4+61 x^3+50 x^2+95 x+93$
• $y^2=48 x^6+12 x^5+106 x^4+81 x^3+70 x^2+109 x+49$
• $y^2=79 x^6+44 x^5+76 x^4+36 x^3+8 x^2+28$
• $y^2=60 x^6+21 x^5+63 x^3+100 x^2+30 x+112$
• $y^2=76 x^6+103 x^5+51 x^4+95 x^3+29 x^2+78 x+66$
• $y^2=78 x^6+13 x^5+4 x^4+52 x^3+17 x^2+55 x+110$
• $y^2=41 x^6+21 x^5+65 x^4+40 x^3+74 x^2+111 x+62$
• $y^2=94 x^6+67 x^5+71 x^4+97 x^3+102 x^2+8 x+38$
• $y^2=36 x^6+99 x^5+7 x^4+70 x^3+76 x^2+66 x+58$
• $y^2=8 x^6+18 x^5+99 x^4+102 x^3+42 x^2+18 x+31$
• $y^2=51 x^6+105 x^5+6 x^4+77 x^3+97 x^2+106 x+48$
• $y^2=78 x^6+33 x^5+12 x^4+58 x^3+2 x^2+17 x+27$
• $y^2=98 x^6+74 x^5+112 x^4+104 x^3+99 x^2+60 x+20$
• $y^2=41 x^6+57 x^5+33 x^4+82 x^3+19 x^2+58 x+70$
• $y^2=4 x^6+107 x^5+101 x^4+57 x^3+111 x^2+26 x+108$
• $y^2=53 x^6+60 x^5+41 x^4+44 x^3+72 x^2+58 x+66$
• $y^2=86 x^6+13 x^5+4 x^4+7 x^3+13 x^2+90 x+70$
• $y^2=101 x^6+27 x^5+65 x^4+106 x^3+57 x^2+37 x+38$
• $y^2=12 x^6+20 x^5+33 x^4+89 x^3+44 x^2+30 x+46$
• $y^2=38 x^6+10 x^5+28 x^4+108 x^3+28 x^2+20 x+43$
• $y^2=96 x^6+107 x^5+27 x^4+42 x^3+76 x^2+62 x+50$
• $y^2=63 x^6+17 x^5+59 x^4+26 x^3+37 x^2+60 x+38$
• $y^2=74 x^6+11 x^5+65 x^4+90 x^3+79 x^2+26 x+92$
• $y^2=40 x^6+14 x^5+37 x^4+30 x^3+9 x^2+106 x$
• $y^2=10 x^6+65 x^5+101 x^4+90 x^3+45 x^2+69 x+96$
• $y^2=14 x^6+91 x^5+54 x^4+3 x^3+77 x^2+29 x+26$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{113}$

The endomorphism algebra of this simple isogeny class is 4.0.535296256.1.

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.y_nk	$2$	(not in LMFDB)