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

• Label: 2.113.ax_ki
• 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:	$( 1 - 21 x + 113 x^{2} )( 1 - 2 x + 113 x^{2} )$
	$1 - 23 x + 268 x^{2} - 2599 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.0498602789898$, $\pm0.470011582631$
Angle rank:	$2$ (numerical)
Jacobians:	$80$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10416$	$163114560$	$2079829257408$	$26577198062956800$	$339450924640560536496$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$91$	$12777$	$1441426$	$163002929$	$18424036211$	$2081952487134$	$235260552829715$	$26584441589429281$	$3004041934829017138$	$339456739013730540057$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 80 curves (of which all are hyperelliptic):

• $y^2=72 x^6+98 x^5+73 x^4+48 x^3+53 x^2+72 x+35$
• $y^2=111 x^6+28 x^5+80 x^4+14 x^3+39 x^2+59 x+75$
• $y^2=87 x^6+42 x^5+99 x^4+6 x^3+92 x^2+69 x+19$
• $y^2=37 x^6+63 x^5+76 x^4+81 x^3+47 x^2+57 x+15$
• $y^2=88 x^6+7 x^4+49 x^3+48 x^2+62 x+31$
• $y^2=72 x^6+41 x^5+18 x^4+52 x^3+82 x^2+89 x+75$
• $y^2=51 x^6+7 x^5+87 x^4+45 x^3+80 x^2+86 x+99$
• $y^2=38 x^6+53 x^5+12 x^4+4 x^3+15 x^2+7 x+87$
• $y^2=89 x^6+69 x^5+85 x^4+82 x^3+93 x^2+6 x+37$
• $y^2=87 x^6+19 x^5+30 x^4+8 x^3+67 x^2+91 x+43$
• $y^2=24 x^6+10 x^5+83 x^4+38 x^3+79 x^2+73 x+102$
• $y^2=46 x^6+17 x^5+65 x^4+10 x^3+14 x^2+78 x$
• $y^2=65 x^6+13 x^5+99 x^4+5 x^3+41 x^2+14 x+74$
• $y^2=39 x^6+50 x^5+42 x^4+106 x^3+86 x^2+75 x+97$
• $y^2=31 x^6+109 x^5+75 x^4+9 x^3+9 x^2+67 x+39$
• $y^2=29 x^6+11 x^5+39 x^4+77 x^3+65 x^2+22 x+3$
• $y^2=37 x^6+80 x^5+31 x^4+70 x^3+99 x^2+84 x+86$
• $y^2=29 x^6+89 x^5+64 x^4+75 x^3+52 x^2+6 x+34$
• $y^2=39 x^6+88 x^5+34 x^4+82 x^3+59 x^2+48 x+15$
• $y^2=19 x^6+83 x^5+109 x^4+99 x^3+30 x^2+4 x+7$
• and

• $y^2=23 x^6+89 x^5+47 x^4+34 x^3+26 x^2+66 x+38$
• $y^2=48 x^6+65 x^5+69 x^4+90 x^3+45 x^2+30 x+105$
• $y^2=99 x^6+69 x^4+56 x^3+2 x^2+87 x+48$
• $y^2=90 x^6+48 x^5+24 x^4+65 x^3+57 x^2+88 x+111$
• $y^2=12 x^6+52 x^5+59 x^4+44 x^3+83 x^2+102 x+63$
• $y^2=83 x^6+110 x^5+92 x^4+52 x^3+16 x^2+10 x+9$
• $y^2=107 x^6+108 x^5+88 x^4+81 x^3+72 x^2+81 x+95$
• $y^2=38 x^6+59 x^5+93 x^4+73 x^3+76 x^2+60 x+77$
• $y^2=50 x^6+73 x^5+104 x^4+60 x^3+95 x^2+103 x+33$
• $y^2=103 x^6+59 x^5+72 x^4+58 x^3+33 x^2+78 x+67$
• $y^2=34 x^6+100 x^5+39 x^4+51 x^3+26 x^2+104 x+52$
• $y^2=2 x^6+84 x^5+102 x^4+52 x^3+45 x^2+72 x+65$
• $y^2=25 x^6+13 x^5+44 x^4+85 x^3+39 x^2+98 x+15$
• $y^2=100 x^6+38 x^5+84 x^4+57 x^3+70 x^2+52 x+86$
• $y^2=78 x^6+62 x^5+11 x^4+24 x^3+104 x^2+4 x+68$
• $y^2=10 x^6+x^5+96 x^4+22 x^3+87 x^2+57 x+19$
• $y^2=70 x^6+91 x^5+63 x^3+53 x^2+20 x+57$
• $y^2=67 x^6+61 x^5+37 x^4+76 x^3+55 x^2+31 x+70$
• $y^2=84 x^6+62 x^5+20 x^4+44 x^3+78 x^2+90 x+100$
• $y^2=101 x^6+94 x^5+14 x^3+77 x^2+10 x+53$
• $y^2=64 x^6+31 x^5+81 x^4+99 x^3+27 x^2+66 x+1$
• $y^2=78 x^6+96 x^5+100 x^4+55 x^3+12 x^2+58 x+94$
• $y^2=83 x^6+8 x^5+77 x^4+81 x^3+85 x^2+65 x+84$
• $y^2=47 x^6+81 x^5+22 x^4+32 x^3+42 x^2+91 x$
• $y^2=23 x^6+23 x^5+66 x^4+12 x^3+25 x^2+26 x+60$
• $y^2=21 x^6+87 x^5+70 x^4+107 x^3+34 x^2+8 x+20$
• $y^2=85 x^6+73 x^5+57 x^4+36 x^3+101 x^2+3 x+8$
• $y^2=87 x^6+89 x^5+39 x^4+112 x^2+102 x+90$
• $y^2=42 x^6+32 x^5+83 x^4+105 x^3+79 x^2+99 x+26$
• $y^2=107 x^6+x^5+87 x^4+71 x^3+95 x^2+94 x+9$
• $y^2=93 x^6+92 x^5+81 x^4+30 x^3+36 x^2+62 x+103$
• $y^2=21 x^6+30 x^5+99 x^4+112 x^3+31 x^2+72 x+38$
• $y^2=54 x^6+23 x^5+24 x^4+98 x^3+82 x^2+34 x+59$
• $y^2=111 x^6+58 x^5+83 x^4+64 x^3+53 x^2+93 x+78$
• $y^2=67 x^6+18 x^5+40 x^4+31 x^3+102 x^2+30 x+51$
• $y^2=45 x^6+29 x^5+108 x^4+7 x^3+18 x^2+22 x+100$
• $y^2=38 x^6+39 x^5+22 x^4+81 x^3+72 x^2+52 x+87$
• $y^2=53 x^6+112 x^5+45 x^4+7 x^3+53 x^2+47 x+112$
• $y^2=5 x^6+15 x^5+81 x^4+29 x^2+108 x+16$
• $y^2=19 x^6+30 x^5+17 x^4+102 x^3+30 x^2+111 x+70$
• $y^2=47 x^6+93 x^5+x^4+108 x^3+17 x^2+68 x+13$
• $y^2=111 x^6+23 x^5+65 x^4+8 x^3+51 x^2+61 x+8$
• $y^2=102 x^6+38 x^5+18 x^4+36 x^3+75 x^2+51 x+9$
• $y^2=57 x^6+22 x^5+78 x^4+97 x^3+5 x^2+49 x+94$
• $y^2=35 x^6+76 x^5+25 x^4+10 x^3+100 x^2+22 x+53$
• $y^2=63 x^6+34 x^5+22 x^4+4 x^2+86 x+85$
• $y^2=45 x^6+51 x^5+51 x^4+111 x^3+50 x^2+21 x+58$
• $y^2=75 x^6+7 x^5+23 x^4+38 x^3+68 x^2+35 x+33$
• $y^2=94 x^6+108 x^5+42 x^4+112 x^3+70 x^2+67 x+38$
• $y^2=92 x^6+79 x^5+3 x^4+90 x^3+83 x^2+34 x+59$
• $y^2=59 x^6+22 x^5+43 x^4+x^3+111 x^2+39 x+4$
• $y^2=48 x^6+101 x^5+33 x^4+54 x^3+56 x^2+42 x+54$
• $y^2=40 x^6+22 x^5+52 x^4+19 x^3+65 x^2+112 x+92$
• $y^2=94 x^6+2 x^5+39 x^4+33 x^3+92 x^2+69 x+29$
• $y^2=83 x^6+29 x^5+14 x^4+81 x^3+94 x^2+67 x+79$
• $y^2=80 x^6+72 x^5+55 x^4+88 x^3+10 x^2+45 x+16$
• $y^2=3 x^6+107 x^5+65 x^4+5 x^3+35 x^2+92 x+22$
• $y^2=24 x^6+94 x^5+41 x^4+75 x^3+20 x^2+17 x+7$
• $y^2=37 x^6+37 x^5+5 x^4+77 x^3+77 x^2+104 x+94$
• $y^2=30 x^6+78 x^5+41 x^4+27 x^3+78 x^2+67 x+104$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{113}$

The isogeny class factors as 1.113.av $\times$ 1.113.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.113.av : \(\Q(\sqrt{-11}) \).
• 1.113.ac : \(\Q(\sqrt{-7}) \).

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.at_hc	$2$	(not in LMFDB)
2.113.t_hc	$2$	(not in LMFDB)
2.113.x_ki	$2$	(not in LMFDB)