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

• Label: 2.113.ax_km
• 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:	$1 - 23 x + 272 x^{2} - 2599 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.0672842588722$, $\pm0.466808076726$
Angle rank:	$2$ (numerical)
Number field:	4.0.66177900.3
Galois group:	$D_{4}$
Jacobians:	$112$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10420$	$163218880$	$2080227501520$	$26577873635526400$	$339451789105489160500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$91$	$12785$	$1441702$	$163007073$	$18424083131$	$2081953232030$	$235260565708427$	$26584441749120193$	$3004041936316622086$	$339456739029828261425$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 112 curves (of which all are hyperelliptic):

• $y^2=9 x^6+26 x^5+85 x^4+41 x^3+77 x^2+17 x+43$
• $y^2=105 x^6+72 x^5+82 x^4+97 x^3+20 x^2+93 x+10$
• $y^2=59 x^6+101 x^5+95 x^4+92 x^3+4 x^2+29 x+29$
• $y^2=21 x^6+25 x^5+72 x^4+81 x^3+6 x^2+14 x+110$
• $y^2=112 x^6+66 x^5+52 x^4+23 x^3+49 x^2+67 x+45$
• $y^2=40 x^6+13 x^5+55 x^4+43 x^3+98 x^2+71 x+85$
• $y^2=26 x^6+51 x^5+69 x^4+33 x^3+28 x^2+75 x+12$
• $y^2=37 x^6+70 x^4+96 x^3+77 x^2+91 x+84$
• $y^2=61 x^6+108 x^5+11 x^4+21 x^3+72 x^2+67 x+23$
• $y^2=59 x^6+10 x^5+24 x^4+65 x^3+6 x^2+2 x+60$
• $y^2=x^6+64 x^5+72 x^4+23 x^3+57 x^2+44 x+12$
• $y^2=65 x^6+52 x^5+80 x^4+100 x^3+25 x^2+66 x+27$
• $y^2=23 x^6+16 x^5+100 x^4+108 x^3+55 x^2+91 x+73$
• $y^2=100 x^6+110 x^5+66 x^4+51 x^3+61 x^2+90 x+12$
• $y^2=46 x^6+84 x^5+60 x^4+29 x^3+82 x^2+x+12$
• $y^2=91 x^6+76 x^5+104 x^4+3 x^3+105 x^2+63 x+73$
• $y^2=58 x^6+91 x^5+85 x^4+89 x^3+56 x^2+86 x+36$
• $y^2=48 x^6+13 x^5+48 x^4+5 x^3+15 x^2+36 x+44$
• $y^2=x^6+92 x^5+107 x^4+45 x^3+44 x^2+62 x+66$
• $y^2=94 x^6+81 x^5+36 x^4+41 x^3+45 x^2+38 x+68$
• and

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

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 4.0.66177900.3.

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.x_km	$2$	(not in LMFDB)