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

• Label: 2.113.ax_kz
• 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:	$1 - 23 x + 285 x^{2} - 2599 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.107532636921$, $\pm0.455819855076$
Angle rank:	$2$ (numerical)
Number field:	4.0.1889794037.1
Galois group:	$D_{4}$
Jacobians:	$81$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10433$	$163558141$	$2081521977209$	$26579997215609141$	$339454130356384853248$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$91$	$12811$	$1442599$	$163020099$	$18424210206$	$2081954965411$	$235260595253215$	$26584442079343203$	$3004041938144296363$	$339456739034743847646$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 81 curves (of which all are hyperelliptic):

• $y^2=78 x^6+67 x^5+60 x^4+83 x^3+x^2+87 x+31$
• $y^2=61 x^6+99 x^5+84 x^4+104 x^3+98 x^2+15 x+106$
• $y^2=80 x^6+105 x^5+62 x^4+50 x^3+100 x^2+90 x+97$
• $y^2=61 x^6+100 x^5+82 x^4+81 x^3+82 x^2+42 x+94$
• $y^2=90 x^6+60 x^5+52 x^4+46 x^3+95 x^2+27 x+36$
• $y^2=31 x^6+45 x^5+6 x^4+103 x^3+95 x^2+69 x+68$
• $y^2=5 x^6+82 x^5+37 x^4+60 x^3+82 x^2+97 x+56$
• $y^2=96 x^6+58 x^5+22 x^4+42 x^3+104 x^2+84 x+33$
• $y^2=61 x^6+81 x^5+19 x^4+60 x^3+26 x^2+x+85$
• $y^2=40 x^6+75 x^5+33 x^4+98 x^3+101 x^2+79 x+56$
• $y^2=71 x^6+22 x^5+85 x^4+51 x^3+96 x^2+42 x+58$
• $y^2=54 x^6+103 x^5+7 x^4+72 x^3+5 x^2+40 x+10$
• $y^2=34 x^6+22 x^5+79 x^4+111 x^3+51 x^2+26 x+95$
• $y^2=74 x^6+109 x^5+11 x^4+75 x^3+50 x^2+90 x+73$
• $y^2=x^6+23 x^5+69 x^4+30 x^3+11 x^2+27 x+38$
• $y^2=42 x^6+76 x^5+86 x^4+27 x^3+21 x^2+10 x+92$
• $y^2=24 x^6+6 x^5+63 x^4+79 x^3+78 x^2+18 x+71$
• $y^2=105 x^6+92 x^5+105 x^4+100 x^3+109 x^2+10 x+76$
• $y^2=66 x^6+70 x^5+32 x^4+41 x^3+26 x^2+75 x+72$
• $y^2=32 x^6+82 x^5+52 x^4+44 x^3+28 x^2+94 x+107$
• and

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

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 4.0.1889794037.1.

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_kz	$2$	(not in LMFDB)