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

• Label: 2.113.ay_ng
• 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:	$1 - 24 x + 344 x^{2} - 2712 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.202556468854$, $\pm0.394772061326$
Angle rank:	$2$ (numerical)
Number field:	4.0.698151168.1
Galois group:	$D_{4}$
Jacobians:	$136$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10378$	$164491300$	$2086007899018$	$26586863701866000$	$339456751779030031498$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$90$	$12882$	$1445706$	$163062214$	$18424352490$	$2081952845394$	$235260578455482$	$26584442095244926$	$3004041934466077338$	$339456738919139566482$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 136 curves (of which all are hyperelliptic):

• $y^2=66 x^6+20 x^5+42 x^4+16 x^3+12 x^2+3 x+110$
• $y^2=84 x^6+75 x^5+112 x^4+6 x^3+19 x^2+48 x+75$
• $y^2=94 x^6+43 x^5+43 x^4+111 x^3+53 x^2+45 x+42$
• $y^2=94 x^6+34 x^5+52 x^4+62 x^3+26 x^2+4 x+84$
• $y^2=46 x^6+77 x^5+46 x^4+72 x^3+72 x^2+39 x+97$
• $y^2=94 x^6+52 x^5+99 x^4+50 x^3+111 x^2+16 x+18$
• $y^2=5 x^6+61 x^5+14 x^4+80 x^3+86 x^2+81 x+14$
• $y^2=91 x^6+94 x^5+37 x^4+54 x^3+26 x^2+24 x+12$
• $y^2=108 x^6+30 x^5+51 x^4+104 x^3+39 x^2+110 x+40$
• $y^2=48 x^6+63 x^5+88 x^4+111 x^3+66 x^2+94 x+1$
• $y^2=17 x^6+17 x^5+103 x^4+4 x^3+28 x^2+89 x+75$
• $y^2=102 x^6+92 x^5+66 x^4+11 x^3+28 x^2+68 x+62$
• $y^2=85 x^6+85 x^5+85 x^4+66 x^3+58 x^2+11 x+56$
• $y^2=81 x^6+9 x^5+94 x^4+82 x^3+36 x^2+65 x+53$
• $y^2=108 x^6+55 x^4+67 x^3+14 x^2+27 x+74$
• $y^2=23 x^6+105 x^5+26 x^4+21 x^3+32 x^2+107 x+88$
• $y^2=72 x^6+20 x^5+37 x^4+14 x^3+67 x^2+5 x+96$
• $y^2=110 x^6+65 x^5+47 x^4+78 x^3+57 x^2+53 x+80$
• $y^2=73 x^6+57 x^5+55 x^4+43 x^3+50 x^2+64 x+10$
• $y^2=81 x^6+36 x^5+32 x^4+34 x^3+74 x^2+90 x$
• and

• $y^2=82 x^6+37 x^5+84 x^4+101 x^3+16 x^2+105 x+75$
• $y^2=16 x^6+19 x^5+38 x^4+15 x^3+66 x^2+76 x+47$
• $y^2=96 x^6+110 x^5+112 x^4+33 x^3+32 x^2+72 x+2$
• $y^2=72 x^6+39 x^5+40 x^4+83 x^3+83 x^2+110 x+92$
• $y^2=10 x^6+8 x^5+71 x^4+98 x^3+53 x^2+12 x+34$
• $y^2=110 x^6+90 x^5+78 x^4+14 x^3+100 x^2+88 x+103$
• $y^2=27 x^6+41 x^5+85 x^4+110 x^3+26 x^2+30 x+10$
• $y^2=87 x^5+42 x^4+66 x^3+8 x^2+81 x+98$
• $y^2=40 x^6+33 x^5+33 x^4+101 x^3+31 x^2+65 x+103$
• $y^2=110 x^6+39 x^5+75 x^4+106 x^3+106 x^2+85 x+29$
• $y^2=56 x^6+28 x^5+105 x^4+42 x^3+10 x^2+47 x+110$
• $y^2=14 x^6+28 x^5+103 x^4+89 x^3+17 x^2+84 x+24$
• $y^2=49 x^5+9 x^4+50 x^3+74 x^2+79 x+99$
• $y^2=6 x^6+x^5+112 x^4+65 x^3+111 x^2+105 x+97$
• $y^2=33 x^6+61 x^5+31 x^4+14 x^3+27 x^2+9 x+19$
• $y^2=35 x^6+81 x^5+93 x^4+67 x^3+57 x^2+47 x+112$
• $y^2=88 x^6+103 x^5+28 x^4+34 x^3+90 x^2+10 x+98$
• $y^2=93 x^6+36 x^5+27 x^4+4 x^3+106 x^2+82 x+41$
• $y^2=8 x^6+18 x^5+78 x^4+112 x^3+61 x^2+91 x+37$
• $y^2=49 x^6+84 x^5+6 x^4+64 x^3+8 x^2+29 x+29$
• $y^2=25 x^6+87 x^5+59 x^4+22 x^3+70 x^2+97 x+63$
• $y^2=91 x^6+90 x^5+74 x^4+27 x^3+50 x^2+61 x+68$
• $y^2=47 x^6+7 x^5+107 x^4+79 x^3+54 x^2+107 x+48$
• $y^2=90 x^6+46 x^5+94 x^4+105 x^3+85 x^2+14 x+86$
• $y^2=100 x^6+61 x^5+51 x^4+11 x^3+5 x^2+x+107$
• $y^2=83 x^6+81 x^5+82 x^4+47 x^3+101 x^2+19 x+58$
• $y^2=22 x^6+44 x^5+10 x^4+13 x^3+112 x^2+49 x+22$
• $y^2=63 x^6+91 x^5+63 x^4+54 x^3+31 x^2+15 x+93$
• $y^2=38 x^6+30 x^5+30 x^4+52 x^3+103 x^2+33 x+96$
• $y^2=45 x^6+100 x^5+88 x^4+21 x^3+107 x^2+17 x+8$
• $y^2=79 x^6+81 x^5+86 x^4+58 x^3+7 x^2+59 x+30$
• $y^2=21 x^6+50 x^5+98 x^4+94 x^3+81 x^2+66 x+37$
• $y^2=91 x^6+82 x^5+92 x^4+42 x^3+91 x^2+46 x+4$
• $y^2=60 x^6+25 x^5+94 x^4+105 x^3+108 x^2+8 x+46$
• $y^2=82 x^6+13 x^5+x^4+77 x^3+10 x^2+83 x+85$
• $y^2=45 x^6+63 x^5+55 x^4+102 x^3+7 x^2+88 x+76$
• $y^2=50 x^6+109 x^5+102 x^4+44 x^3+109 x^2+9 x+4$
• $y^2=55 x^6+11 x^5+102 x^4+109 x^3+2 x^2+62 x+73$
• $y^2=76 x^6+47 x^5+91 x^4+96 x^3+107 x^2+45 x+20$
• $y^2=5 x^6+107 x^5+79 x^4+83 x^3+40 x^2+57 x+88$
• $y^2=34 x^6+43 x^5+24 x^4+55 x^3+57 x^2+18 x+93$
• $y^2=50 x^6+96 x^5+106 x^4+62 x^3+65 x^2+17 x+101$
• $y^2=27 x^6+94 x^5+89 x^4+44 x^3+49 x^2+61 x+40$
• $y^2=66 x^6+62 x^5+8 x^4+86 x^3+75 x^2+107 x+103$
• $y^2=108 x^6+86 x^5+42 x^4+108 x^3+98 x^2+33 x+35$
• $y^2=82 x^6+53 x^5+107 x^4+26 x^3+105 x^2+101 x+70$
• $y^2=67 x^6+108 x^5+112 x^4+90 x^3+109 x^2+10 x+24$
• $y^2=110 x^6+16 x^5+32 x^4+60 x^3+108 x^2+96 x+39$
• $y^2=96 x^6+110 x^5+87 x^4+21 x^3+19 x^2+94 x+105$
• $y^2=68 x^6+98 x^5+18 x^4+70 x^3+37 x^2+55 x+36$
• $y^2=9 x^6+20 x^5+7 x^4+107 x^3+105 x^2+57 x+88$
• $y^2=87 x^6+32 x^5+18 x^4+3 x^3+55 x^2+100 x+101$
• $y^2=102 x^6+41 x^5+5 x^4+26 x^3+98 x^2+53 x+17$
• $y^2=90 x^6+x^5+43 x^4+51 x^3+96 x^2+109 x+87$
• $y^2=104 x^6+18 x^5+37 x^4+81 x^3+83 x^2+68 x+95$
• $y^2=3 x^6+91 x^5+107 x^4+63 x^3+8 x^2+x+72$
• $y^2=83 x^6+3 x^5+14 x^4+9 x^3+16 x^2+56 x+39$
• $y^2=112 x^6+16 x^5+80 x^4+75 x^3+86 x^2+110 x+95$
• $y^2=103 x^6+5 x^5+102 x^4+102 x^3+101 x^2+89 x+103$
• $y^2=38 x^6+59 x^5+44 x^4+92 x^3+40 x^2+93 x+19$
• $y^2=33 x^6+25 x^5+32 x^4+14 x^3+80 x^2+68 x+37$
• $y^2=87 x^6+45 x^5+47 x^4+59 x^3+22 x^2+31 x+72$
• $y^2=30 x^6+94 x^5+4 x^4+59 x^3+70 x^2+108 x+31$
• $y^2=54 x^6+15 x^5+104 x^4+15 x^3+11 x^2+28 x+44$
• $y^2=73 x^6+109 x^5+19 x^4+28 x^3+59 x^2+3 x+107$
• $y^2=38 x^6+78 x^5+88 x^4+53 x^3+34 x^2+62 x+73$
• $y^2=65 x^6+62 x^5+98 x^4+46 x^3+34 x^2+97 x+86$
• $y^2=90 x^6+32 x^5+20 x^4+49 x^3+93 x^2+112 x+10$
• $y^2=35 x^6+107 x^5+99 x^4+35 x^3+3 x^2+71 x+33$
• $y^2=23 x^6+29 x^5+25 x^4+69 x^3+19 x^2+90 x+41$
• $y^2=111 x^6+64 x^5+38 x^4+8 x^3+83 x^2+87 x+8$
• $y^2=53 x^6+14 x^5+43 x^4+88 x^3+37 x^2+25 x+74$
• $y^2=108 x^6+106 x^5+38 x^4+83 x^3+94 x^2+50 x+13$
• $y^2=50 x^6+52 x^5+73 x^4+69 x^3+23 x^2+11 x+59$
• $y^2=12 x^6+78 x^5+111 x^4+39 x^3+54 x^2+17 x+92$
• $y^2=59 x^6+9 x^5+23 x^4+84 x^3+90 x^2+98 x+65$
• $y^2=10 x^6+17 x^5+9 x^4+65 x^3+95 x^2+53 x+59$
• $y^2=55 x^5+10 x^4+43 x^3+39 x^2+74 x+94$
• $y^2=5 x^6+25 x^5+67 x^4+82 x^3+36 x^2+86 x+110$
• $y^2=37 x^6+101 x^5+104 x^4+34 x^3+40 x^2+94 x+5$
• $y^2=82 x^6+77 x^5+96 x^4+50 x^3+83 x^2+15 x+17$
• $y^2=90 x^6+x^5+3 x^4+11 x^3+49 x^2+106 x+20$
• $y^2=97 x^6+40 x^5+111 x^4+81 x^3+83 x^2+75 x+39$
• $y^2=10 x^6+58 x^5+107 x^4+18 x^3+52 x^2+10 x+30$
• $y^2=46 x^6+54 x^5+6 x^4+79 x^3+42 x^2+82 x+104$
• $y^2=82 x^6+103 x^5+73 x^4+11 x^3+33 x^2+18 x+107$
• $y^2=108 x^6+83 x^5+111 x^4+44 x^3+51 x^2+102 x+36$
• $y^2=96 x^6+13 x^5+62 x^4+85 x^3+73 x^2+33 x+106$
• $y^2=71 x^6+37 x^5+65 x^4+66 x^3+76 x^2+24 x+11$
• $y^2=62 x^6+36 x^4+94 x^3+46 x^2+105 x+76$
• $y^2=106 x^6+47 x^5+97 x^4+57 x^3+19 x^2+82 x+90$
• $y^2=108 x^6+14 x^5+51 x^4+66 x^3+108 x^2+55 x+89$
• $y^2=75 x^6+59 x^5+53 x^4+47 x^3+2 x^2+57 x+67$
• $y^2=62 x^6+26 x^5+58 x^4+36 x^3+91 x^2+44 x+70$
• $y^2=14 x^6+42 x^5+51 x^4+66 x^3+75 x^2+98 x+18$
• $y^2=89 x^6+3 x^5+27 x^4+73 x^3+94 x^2+64 x+34$
• $y^2=5 x^6+45 x^5+77 x^4+87 x^3+20 x^2+43 x+52$
• $y^2=14 x^6+65 x^5+95 x^4+96 x^3+65 x^2+44 x+77$
• $y^2=55 x^6+34 x^5+5 x^4+106 x^3+75 x^2+24 x+84$
• $y^2=52 x^6+20 x^5+8 x^4+3 x^3+63 x^2+35 x+66$
• $y^2=98 x^6+63 x^5+13 x^4+76 x^3+47 x^2+83 x+52$
• $y^2=17 x^6+89 x^5+66 x^4+73 x^3+102 x^2+64 x+24$
• $y^2=33 x^6+108 x^5+61 x^4+66 x^3+66 x^2+41 x+81$
• $y^2=65 x^6+80 x^5+53 x^4+54 x^3+57 x^2+34 x+28$
• $y^2=10 x^6+12 x^5+75 x^4+3 x^3+2 x^2+85 x+40$
• $y^2=52 x^6+20 x^5+112 x^4+16 x^3+60 x^2+50 x+14$
• $y^2=94 x^6+42 x^5+23 x^4+37 x^3+99 x^2+72 x+51$
• $y^2=25 x^6+108 x^5+48 x^4+99 x^3+73 x^2+50 x+90$
• $y^2=106 x^6+46 x^5+17 x^4+67 x^3+46 x+2$
• $y^2=87 x^6+73 x^5+63 x^4+32 x^3+29 x^2+85 x+80$
• $y^2=86 x^6+61 x^5+61 x^4+88 x^3+46 x^2+78 x+54$
• $y^2=15 x^6+74 x^5+7 x^4+16 x^3+32 x^2+96 x+110$
• $y^2=42 x^6+110 x^5+75 x^4+55 x^3+59 x^2+75 x+70$
• $y^2=85 x^6+97 x^5+92 x^4+26 x^3+37 x^2+31 x+58$
• $y^2=75 x^6+87 x^5+109 x^3+10 x^2+37 x+70$
• $y^2=35 x^6+95 x^5+65 x^4+65 x^3+98 x^2+105 x+12$

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 4.0.698151168.1.

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_ng	$2$	(not in LMFDB)