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

• Label: 2.113.ax_ni
• 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:	$( 1 - 15 x + 113 x^{2} )( 1 - 8 x + 113 x^{2} )$
	$1 - 23 x + 346 x^{2} - 2599 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.250704227710$, $\pm0.377200205714$
Angle rank:	$2$ (numerical)
Jacobians:	$60$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10494$	$165154572$	$2087600131584$	$26588490866175744$	$339455552028792766974$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$91$	$12933$	$1446808$	$163072193$	$18424287371$	$2081949760098$	$235260539116379$	$26584441928616961$	$3004041936707297464$	$339456738965244145893$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 60 curves (of which all are hyperelliptic):

• $y^2=84 x^6+94 x^5+87 x^4+83 x^3+12 x^2+9 x+90$
• $y^2=6 x^6+29 x^4+7 x^3+51 x^2+35 x+50$
• $y^2=89 x^6+82 x^5+15 x^4+45 x^3+4 x^2+18 x+13$
• $y^2=74 x^6+67 x^5+69 x^4+27 x^3+96 x^2+77 x+31$
• $y^2=25 x^6+67 x^5+15 x^4+63 x^3+31 x^2+19 x+35$
• $y^2=42 x^6+80 x^5+11 x^4+16 x^3+25 x^2+41 x+49$
• $y^2=16 x^6+83 x^5+106 x^4+109 x^3+19 x^2+26 x+89$
• $y^2=55 x^6+60 x^5+82 x^4+82 x^3+42 x^2+30 x+38$
• $y^2=21 x^6+46 x^5+82 x^4+9 x^3+72 x^2+104 x+27$
• $y^2=104 x^6+90 x^5+101 x^4+31 x^3+13 x^2+103 x+106$
• $y^2=71 x^6+101 x^5+7 x^4+x^3+82 x^2+30 x+45$
• $y^2=62 x^6+5 x^5+3 x^4+90 x^3+27 x^2+7 x+108$
• $y^2=35 x^6+88 x^5+93 x^4+70 x^3+99 x^2+15 x+68$
• $y^2=67 x^6+60 x^5+75 x^4+76 x^3+88 x^2+50 x+48$
• $y^2=97 x^6+36 x^5+32 x^4+99 x^3+106 x^2+32 x+3$
• $y^2=66 x^6+112 x^5+99 x^4+108 x^3+101 x^2+38 x+3$
• $y^2=39 x^6+14 x^5+73 x^4+5 x^3+97 x^2+107 x+107$
• $y^2=102 x^6+98 x^5+6 x^4+36 x^3+x^2+77 x+75$
• $y^2=5 x^6+4 x^5+82 x^4+92 x^3+87 x^2+57 x+64$
• $y^2=51 x^6+96 x^5+38 x^4+78 x^3+77 x^2+52 x+19$
• and

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{113}$

The isogeny class factors as 1.113.ap $\times$ 1.113.ai 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.ap : \(\Q(\sqrt{-227}) \).
• 1.113.ai : \(\Q(\sqrt{-97}) \).

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.ah_ec	$2$	(not in LMFDB)
2.113.h_ec	$2$	(not in LMFDB)
2.113.x_ni	$2$	(not in LMFDB)