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

• Label: 2.113.ax_mn
• 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:	$1 - 23 x + 325 x^{2} - 2599 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.198303859222$, $\pm0.413078296252$
Angle rank:	$2$ (numerical)
Number field:	4.0.9428237.2
Galois group:	$D_{4}$
Jacobians:	$182$
Isomorphism classes:	182

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10473$	$164604141$	$2085506849169$	$26585840473249941$	$339456842354411144448$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$91$	$12891$	$1445359$	$163055939$	$18424357406$	$2081953945251$	$235260587518775$	$26584442029422883$	$3004041933090493483$	$339456738921643943646$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 182 curves (of which all are hyperelliptic):

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

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

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 4.0.9428237.2.

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_mn	$2$	(not in LMFDB)