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

• Label: 2.113.ay_og
• Base field: $\F_{113}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{113}$
Dimension:	$2$
L-polynomial:	$( 1 - 12 x + 113 x^{2} )^{2}$
	$1 - 24 x + 370 x^{2} - 2712 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.309095034261$, $\pm0.309095034261$
Angle rank:	$1$ (numerical)
Jacobians:	$120$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10404$	$165173904$	$2088712876644$	$26590577755262976$	$339455314754473912164$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$90$	$12934$	$1447578$	$163084990$	$18424274490$	$2081946572998$	$235260494624826$	$26584441873320574$	$3004041943351802394$	$339456739062931767814$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 120 curves (of which all are hyperelliptic):

• $y^2=108 x^6+18 x^5+110 x^4+105 x^3+x^2+112 x+5$
• $y^2=39 x^6+26 x^5+94 x^4+40 x^3+94 x^2+26 x+39$
• $y^2=52 x^6+2 x^5+15 x^4+34 x^3+67 x^2+21 x+29$
• $y^2=53 x^6+89 x^5+101 x^4+2 x^3+31 x^2+71 x+12$
• $y^2=20 x^6+75 x^5+94 x^4+58 x^3+83 x^2+112 x+112$
• $y^2=39 x^6+19 x^5+27 x^4+64 x^3+55 x^2+59 x+24$
• $y^2=76 x^6+33 x^5+92 x^4+68 x^3+105 x^2+74 x+36$
• $y^2=3 x^6+95 x^5+87 x^4+80 x^3+87 x^2+95 x+3$
• $y^2=5 x^6+26 x^5+66 x^4+52 x^3+73 x^2+67 x+55$
• $y^2=33 x^6+69 x^5+59 x^4+17 x^3+8 x^2+67 x+74$
• $y^2=49 x^6+36 x^5+110 x^4+41 x^3+15 x^2+49 x+39$
• $y^2=39 x^6+4 x^5+63 x^4+51 x^3+x^2+81 x+100$
• $y^2=105 x^6+24 x^5+53 x^4+104 x^3+14 x^2+45 x+60$
• $y^2=59 x^6+80 x^5+57 x^4+17 x^3+105 x^2+9 x+94$
• $y^2=56 x^6+54 x^5+85 x^4+90 x^3+85 x^2+54 x+56$
• $y^2=3 x^6+111 x^5+65 x^4+10 x^3+36 x^2+44 x+25$
• $y^2=22 x^6+110 x^5+81 x^4+46 x^3+77 x^2+58 x+30$
• $y^2=59 x^6+13 x^5+38 x^4+96 x^3+17 x^2+26 x+29$
• $y^2=98 x^6+12 x^5+103 x^4+38 x^3+100 x^2+83 x+81$
• $y^2=42 x^6+110 x^5+20 x^4+105 x^3+27 x^2+77 x+50$
• and

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{113}$.

Endomorphism algebra over $\F_{113}$

The isogeny class factors as 1.113.am^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-77}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.113.a_de	$2$	(not in LMFDB)
2.113.y_og	$2$	(not in LMFDB)
2.113.m_bf	$3$	(not in LMFDB)