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

• Label: 2.113.au_ia
• 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:	$1 - 20 x + 208 x^{2} - 2260 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.0616501555321$, $\pm0.512921130228$
Angle rank:	$2$ (numerical)
Number field:	4.0.1683355904.1
Galois group:	$D_{4}$
Jacobians:	$96$
Isomorphism classes:	96

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10698$	$163230084$	$2078636559066$	$26577357981121296$	$339454913167390821018$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$94$	$12786$	$1440598$	$163003910$	$18424252694$	$2081953406322$	$235260532899038$	$26584441613049214$	$3004041939817085374$	$339456739039264087986$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 96 curves (of which all are hyperelliptic):

• $y^2=62 x^6+37 x^5+73 x^4+54 x^3+26 x^2+82 x+78$
• $y^2=74 x^6+57 x^5+81 x^4+51 x^3+50 x^2+65 x+102$
• $y^2=111 x^6+27 x^5+100 x^4+92 x^3+93 x^2+74 x+14$
• $y^2=101 x^6+61 x^5+66 x^4+11 x^3+2 x^2+103 x+54$
• $y^2=45 x^6+99 x^5+35 x^4+93 x^3+108 x^2+67 x+31$
• $y^2=110 x^6+104 x^5+57 x^4+77 x^3+62 x^2+15 x+73$
• $y^2=18 x^6+36 x^5+78 x^4+36 x^3+80 x^2+32 x+88$
• $y^2=43 x^6+85 x^5+26 x^4+73 x^3+15 x^2+83 x+12$
• $y^2=47 x^6+105 x^5+72 x^4+38 x^3+2 x^2+56 x+36$
• $y^2=94 x^6+33 x^5+84 x^4+16 x^3+81 x^2+53 x+87$
• $y^2=63 x^6+70 x^5+32 x^4+95 x^3+23 x^2+29 x+71$
• $y^2=101 x^6+76 x^5+19 x^4+38 x^3+99 x^2+91 x+31$
• $y^2=95 x^6+97 x^5+2 x^4+83 x^3+68 x^2+94 x+61$
• $y^2=74 x^6+90 x^5+36 x^4+101 x^3+76 x^2+8$
• $y^2=76 x^6+103 x^5+41 x^4+37 x^3+23 x^2+57 x+2$
• $y^2=100 x^6+34 x^5+81 x^4+33 x^3+2 x^2+26 x+111$
• $y^2=103 x^6+108 x^5+57 x^4+4 x^3+63 x^2+32 x$
• $y^2=57 x^6+58 x^5+68 x^4+79 x^3+10 x^2+87 x+49$
• $y^2=10 x^6+48 x^5+99 x^4+17 x^3+52 x^2+54 x+12$
• $y^2=29 x^6+78 x^5+50 x^4+46 x^3+100 x^2+64 x+7$
• and

• $y^2=83 x^6+35 x^5+60 x^4+71 x^3+38 x^2+56 x+54$
• $y^2=34 x^6+20 x^5+70 x^4+90 x^3+90 x^2+92 x+27$
• $y^2=79 x^6+30 x^5+86 x^4+37 x^3+53 x^2+99 x+79$
• $y^2=82 x^6+75 x^5+14 x^4+72 x^3+108 x^2+81 x+86$
• $y^2=49 x^6+43 x^5+33 x^4+100 x^3+90 x^2+91 x+24$
• $y^2=15 x^6+112 x^5+33 x^4+19 x^3+96 x^2+48 x+101$
• $y^2=81 x^6+101 x^5+103 x^4+49 x^3+20 x^2+111 x+31$
• $y^2=22 x^6+36 x^5+74 x^4+102 x^3+111 x^2+72 x+1$
• $y^2=56 x^6+18 x^5+97 x^4+49 x^3+87 x^2+33 x+80$
• $y^2=92 x^6+67 x^5+65 x^4+4 x^3+52 x^2+95 x+33$
• $y^2=86 x^6+74 x^5+97 x^4+68 x^3+12 x^2+84 x+109$
• $y^2=34 x^6+63 x^5+19 x^4+110 x^3+64 x^2+57 x+43$
• $y^2=101 x^6+110 x^5+4 x^4+54 x^3+97 x^2+13 x+96$
• $y^2=112 x^6+68 x^5+92 x^4+10 x^3+92 x^2+12 x+37$
• $y^2=42 x^6+92 x^5+102 x^4+26 x^3+20 x^2+14 x+103$
• $y^2=68 x^6+19 x^5+79 x^4+13 x^3+60 x^2+82 x+91$
• $y^2=80 x^6+79 x^5+73 x^4+2 x^3+98 x^2+75 x+104$
• $y^2=79 x^6+28 x^5+99 x^4+85 x^3+86 x^2+12 x+59$
• $y^2=92 x^6+16 x^5+87 x^4+10 x^3+23 x^2+103 x+103$
• $y^2=108 x^6+112 x^5+3 x^4+92 x^3+37 x^2+23 x+110$
• $y^2=19 x^6+15 x^5+106 x^4+21 x^3+99 x^2+102 x+25$
• $y^2=3 x^6+66 x^5+67 x^4+92 x^3+50 x^2+25 x+92$
• $y^2=105 x^6+62 x^5+9 x^4+67 x^3+21 x^2+53 x+97$
• $y^2=39 x^6+32 x^5+105 x^4+29 x^3+99 x^2+79 x+65$
• $y^2=58 x^6+93 x^5+35 x^4+49 x^3+22 x^2+34 x+11$
• $y^2=11 x^6+39 x^5+35 x^4+103 x^3+76 x^2+6 x+20$
• $y^2=2 x^6+3 x^5+32 x^4+66 x^3+47 x^2+111 x+112$
• $y^2=97 x^6+5 x^5+41 x^4+49 x^3+95 x^2+39 x+43$
• $y^2=104 x^6+23 x^5+43 x^4+108 x^3+70 x^2+86 x+45$
• $y^2=54 x^6+62 x^5+87 x^4+89 x^3+26 x^2+99 x+84$
• $y^2=7 x^6+83 x^5+31 x^4+8 x^3+16 x^2+62 x+1$
• $y^2=86 x^6+90 x^5+23 x^4+56 x+3$
• $y^2=29 x^6+18 x^5+18 x^4+100 x^3+74 x^2+52 x+58$
• $y^2=77 x^6+36 x^5+98 x^4+20 x^3+99 x^2+107 x+70$
• $y^2=92 x^6+41 x^5+11 x^4+19 x^3+82 x^2+79 x+3$
• $y^2=79 x^6+60 x^5+26 x^4+8 x^3+71 x^2+47 x+105$
• $y^2=71 x^6+84 x^5+24 x^4+36 x^3+85 x^2+43 x+33$
• $y^2=90 x^6+28 x^5+14 x^4+8 x^3+37 x^2+109 x+56$
• $y^2=5 x^6+105 x^5+93 x^4+9 x^3+101 x^2+23 x+2$
• $y^2=22 x^6+84 x^5+99 x^4+76 x^3+83 x^2+71 x+95$
• $y^2=80 x^6+75 x^5+47 x^4+55 x^3+79 x^2+55 x+43$
• $y^2=13 x^6+53 x^5+19 x^4+86 x^3+18 x^2+6 x+37$
• $y^2=92 x^6+36 x^5+42 x^4+16 x^3+31 x^2+76 x+47$
• $y^2=19 x^6+4 x^5+23 x^4+51 x^3+32 x^2+110 x+103$
• $y^2=106 x^6+90 x^5+47 x^4+103 x^3+16 x^2+95 x+93$
• $y^2=69 x^6+21 x^5+91 x^4+42 x^3+78 x^2+17 x+17$
• $y^2=88 x^6+110 x^5+31 x^4+61 x^3+49 x^2+13 x+112$
• $y^2=54 x^6+61 x^5+63 x^4+45 x^3+13 x^2+73 x+27$
• $y^2=89 x^6+30 x^5+97 x^4+63 x^3+56 x^2+29 x+35$
• $y^2=69 x^6+71 x^5+7 x^4+61 x^3+33 x^2+47 x+43$
• $y^2=83 x^6+77 x^5+74 x^4+83 x^3+13 x^2+45 x+37$
• $y^2=40 x^6+10 x^5+106 x^4+61 x^3+22 x^2+111 x+105$
• $y^2=72 x^6+42 x^5+86 x^4+65 x^3+31 x^2+19 x+12$
• $y^2=83 x^6+77 x^5+66 x^4+88 x^3+19 x^2+6 x+101$
• $y^2=84 x^6+13 x^5+68 x^4+90 x^3+13 x^2+33 x+107$
• $y^2=65 x^6+53 x^5+32 x^4+106 x^3+34 x^2+11 x+66$
• $y^2=70 x^6+29 x^5+84 x^4+26 x^3+105 x^2+9 x+25$
• $y^2=51 x^6+57 x^5+84 x^4+73 x^3+65 x^2+23 x+104$
• $y^2=30 x^6+48 x^5+35 x^4+65 x^3+57 x^2+85 x+55$
• $y^2=82 x^6+89 x^5+80 x^4+50 x^3+34 x^2+19 x+78$
• $y^2=11 x^6+70 x^5+104 x^4+73 x^3+111 x^2+56 x+54$
• $y^2=11 x^6+96 x^4+84 x^3+58 x^2+28 x+58$
• $y^2=57 x^6+107 x^5+99 x^4+78 x^3+20 x^2+16 x+82$
• $y^2=92 x^6+88 x^5+39 x^4+46 x^3+15 x^2+3 x+32$
• $y^2=31 x^6+112 x^5+16 x^4+52 x^3+110 x^2+16 x+103$
• $y^2=105 x^6+106 x^5+75 x^4+37 x^3+59 x^2+36 x+22$
• $y^2=95 x^6+88 x^5+96 x^4+79 x^3+29 x^2+18 x+81$
• $y^2=107 x^6+4 x^5+41 x^4+25 x^3+79 x^2+19 x+35$
• $y^2=90 x^6+59 x^5+62 x^4+94 x^3+98 x^2+57 x+56$
• $y^2=58 x^6+50 x^5+16 x^4+97 x^3+58 x^2+76 x+25$
• $y^2=79 x^6+24 x^5+65 x^4+20 x^3+60 x^2+49 x+48$
• $y^2=61 x^6+74 x^5+5 x^4+108 x^3+15 x^2+76 x+16$
• $y^2=60 x^6+2 x^5+58 x^4+35 x^3+11 x^2+107 x+89$
• $y^2=43 x^6+74 x^5+16 x^4+92 x^3+21 x^2+52 x+80$
• $y^2=108 x^6+48 x^5+38 x^4+58 x^3+6 x^2+79 x+2$
• $y^2=103 x^6+34 x^5+17 x^4+12 x^3+102 x^2+48 x+104$

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 4.0.1683355904.1.

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.u_ia	$2$	(not in LMFDB)