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

• Label: 2.113.ax_nq
• 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:	$1 - 23 x + 354 x^{2} - 2599 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.279810831014$, $\pm0.353577770383$
Angle rank:	$2$ (numerical)
Number field:	4.0.28117388.1
Galois group:	$D_{4}$
Jacobians:	$36$
Isomorphism classes:	36

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10502$	$165364492$	$2088397810304$	$26589424980921344$	$339454568936505850822$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$91$	$12949$	$1447360$	$163077921$	$18424234011$	$2081947536802$	$235260515223979$	$26584441970255169$	$3004041941097320896$	$339456739017771257109$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 36 curves (of which all are hyperelliptic):

• $y^2=79 x^6+57 x^5+28 x^4+59 x^3+51 x^2+109 x+98$
• $y^2=3 x^6+51 x^5+72 x^4+61 x^3+81 x^2+78 x+94$
• $y^2=96 x^6+97 x^5+69 x^4+106 x^3+80 x^2+104 x+92$
• $y^2=5 x^6+6 x^5+93 x^4+36 x^3+97 x^2+86 x+62$
• $y^2=68 x^6+33 x^5+59 x^4+79 x^3+72 x^2+54 x+103$
• $y^2=3 x^6+96 x^5+56 x^4+93 x^3+63 x^2+70 x+34$
• $y^2=17 x^6+21 x^5+75 x^4+11 x^3+17 x^2+66 x+85$
• $y^2=103 x^6+83 x^5+24 x^4+105 x^3+49 x^2+5 x+94$
• $y^2=90 x^6+49 x^5+20 x^4+49 x^3+55 x^2+73 x+45$
• $y^2=5 x^6+98 x^5+95 x^4+59 x^3+92 x^2+65 x+53$
• $y^2=49 x^6+69 x^5+10 x^4+x^3+44 x^2+30 x+93$
• $y^2=68 x^6+71 x^5+70 x^4+7 x^3+27 x^2+24 x+92$
• $y^2=80 x^6+51 x^5+69 x^4+39 x^3+92 x^2+33 x+69$
• $y^2=19 x^6+34 x^5+13 x^4+49 x^3+50 x^2+27 x+82$
• $y^2=94 x^6+45 x^5+58 x^4+111 x^3+61 x^2+59 x+9$
• $y^2=110 x^6+28 x^5+80 x^4+110 x^3+102 x^2+7 x+11$
• $y^2=13 x^6+13 x^5+111 x^4+5 x^3+56 x^2+64 x+38$
• $y^2=101 x^6+58 x^5+34 x^4+60 x^3+46 x^2+5 x+111$
• $y^2=60 x^6+10 x^5+37 x^4+26 x^3+3 x^2+5 x+38$
• $y^2=61 x^6+42 x^5+28 x^4+7 x^3+41 x^2+26 x+11$
• and

• $y^2=20 x^6+17 x^5+4 x^4+43 x^3+53 x^2+53 x+81$
• $y^2=63 x^6+96 x^5+42 x^4+24 x^3+109 x^2+97 x+20$
• $y^2=31 x^6+51 x^5+63 x^4+26 x^3+21 x^2+4 x+11$
• $y^2=107 x^5+26 x^4+111 x^3+80 x^2+80 x+5$
• $y^2=40 x^6+53 x^5+76 x^4+30 x^3+55 x^2+13 x+107$
• $y^2=18 x^6+112 x^5+94 x^4+98 x^3+68 x^2+93 x+39$
• $y^2=99 x^6+31 x^5+87 x^4+51 x^3+72 x^2+71 x+1$
• $y^2=39 x^6+104 x^5+106 x^4+85 x^3+92 x^2+14 x+74$
• $y^2=43 x^6+111 x^5+64 x^4+48 x^3+83 x^2+90 x+37$
• $y^2=104 x^6+25 x^5+100 x^4+16 x^3+56 x^2+11 x+66$
• $y^2=97 x^6+57 x^5+66 x^4+103 x^3+88 x^2+36 x+2$
• $y^2=17 x^6+8 x^5+99 x^4+49 x^3+20 x^2+67 x+40$
• $y^2=62 x^6+86 x^5+29 x^4+89 x^3+18 x^2+17 x+55$
• $y^2=88 x^6+109 x^5+63 x^4+2 x^2+46 x+104$
• $y^2=55 x^6+26 x^5+17 x^4+104 x^3+42 x^2+81 x+73$
• $y^2=92 x^6+24 x^5+88 x^4+35 x^3+50 x^2+66 x+93$

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 4.0.28117388.1.

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_nq	$2$	(not in LMFDB)