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

• Label: 2.113.ax_md
• 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:	$1 - 23 x + 315 x^{2} - 2599 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.176453268058$, $\pm0.425609442072$
Angle rank:	$2$ (numerical)
Number field:	4.0.9487493.1
Galois group:	$D_{4}$
Jacobians:	$91$
Isomorphism classes:	91

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10463$	$164342341$	$2084510353679$	$26584477403057141$	$339456799987295761648$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$91$	$12871$	$1444669$	$163047579$	$18424355106$	$2081955073591$	$235260601334185$	$26584442114447763$	$3004041934050875503$	$339456738939451955646$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 91 curves (of which all are hyperelliptic):

• $y^2=107 x^6+99 x^5+88 x^4+68 x^3+90 x^2+93 x+65$
• $y^2=24 x^6+48 x^5+90 x^4+66 x^3+3 x^2+58 x+84$
• $y^2=89 x^6+106 x^5+16 x^4+83 x^3+85 x^2+91 x+22$
• $y^2=104 x^6+97 x^5+35 x^4+30 x^3+76 x^2+47 x+107$
• $y^2=70 x^6+5 x^5+34 x^4+95 x^3+82 x^2+67 x+65$
• $y^2=63 x^6+104 x^5+44 x^4+63 x^3+62 x^2+13 x+79$
• $y^2=23 x^6+110 x^5+103 x^4+21 x^3+87 x^2+64 x+48$
• $y^2=78 x^6+22 x^5+36 x^4+89 x^3+78 x^2+93 x+90$
• $y^2=70 x^6+78 x^5+103 x^4+x^3+98 x^2+34 x+17$
• $y^2=105 x^6+26 x^5+100 x^4+8 x^3+50 x^2+5 x+3$
• $y^2=103 x^6+64 x^5+67 x^4+86 x^3+51 x^2+12 x+75$
• $y^2=34 x^6+97 x^5+44 x^4+3 x^3+101 x^2+26 x+20$
• $y^2=103 x^6+18 x^5+22 x^4+44 x^3+23 x^2+9 x+13$
• $y^2=110 x^6+82 x^5+89 x^4+48 x^3+97 x^2+61 x+47$
• $y^2=9 x^6+89 x^5+51 x^4+81 x^3+18 x^2+19 x+33$
• $y^2=22 x^6+75 x^5+x^4+11 x^3+15 x^2+97 x+102$
• $y^2=112 x^6+77 x^5+49 x^4+111 x^3+33 x^2+72 x+74$
• $y^2=33 x^6+65 x^5+102 x^4+40 x^3+74 x^2+96 x+62$
• $y^2=89 x^6+44 x^5+14 x^4+41 x^3+24 x^2+x+92$
• $y^2=77 x^6+22 x^5+21 x^4+59 x^3+95 x^2+93 x+22$
• and

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

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 4.0.9487493.1.

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.x_md	$2$	(not in LMFDB)