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

• Label: 2.113.az_nz
• 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:	$1 - 25 x + 363 x^{2} - 2825 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.207827179752$, $\pm0.375379823670$
Angle rank:	$2$ (numerical)
Number field:	4.0.381952109.1
Galois group:	$D_{4}$
Jacobians:	$96$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10283$	$164353189$	$2086465708691$	$26588015109316661$	$339457070599663966768$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$89$	$12871$	$1446023$	$163069275$	$18424369794$	$2081951786647$	$235260564362843$	$26584442095702419$	$3004041936093090149$	$339456738930117034686$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 96 curves (of which all are hyperelliptic):

• $y^2=80 x^6+89 x^5+93 x^4+53 x^3+58 x^2+96 x+39$
• $y^2=20 x^6+45 x^5+109 x^4+81 x^3+89 x^2+71 x+39$
• $y^2=101 x^6+54 x^5+109 x^4+10 x^3+98 x^2+107 x+42$
• $y^2=59 x^6+106 x^5+29 x^4+50 x^3+34 x^2+32 x+67$
• $y^2=58 x^6+35 x^5+21 x^4+15 x^3+72 x^2+79 x+101$
• $y^2=65 x^6+33 x^5+19 x^4+51 x^3+70 x^2+17 x+107$
• $y^2=89 x^6+94 x^5+47 x^4+14 x^3+111 x^2+12 x+26$
• $y^2=7 x^6+11 x^5+28 x^4+7 x^3+31 x^2+47 x+43$
• $y^2=92 x^6+80 x^5+87 x^4+5 x^3+83 x^2+4 x+68$
• $y^2=25 x^5+24 x^4+110 x^3+5 x^2+50 x+44$
• $y^2=39 x^6+91 x^5+34 x^4+55 x^3+12 x^2+9 x+2$
• $y^2=17 x^6+92 x^5+82 x^4+38 x^3+76 x^2+92 x+55$
• $y^2=21 x^6+7 x^5+101 x^4+91 x^3+39 x^2+54 x+104$
• $y^2=76 x^6+13 x^5+100 x^4+99 x^3+105 x^2+67 x+57$
• $y^2=35 x^6+16 x^5+39 x^4+3 x^3+63 x^2+57 x+30$
• $y^2=97 x^6+37 x^5+108 x^4+x^3+52 x^2+32 x$
• $y^2=68 x^6+73 x^5+58 x^4+54 x^3+96 x^2+94 x+7$
• $y^2=89 x^6+74 x^5+7 x^4+46 x^3+34 x^2+46 x+109$
• $y^2=41 x^5+46 x^4+34 x^3+65 x^2+69 x+6$
• $y^2=21 x^6+68 x^5+68 x^4+110 x^3+15 x^2+27 x+52$
• and

• $y^2=39 x^6+4 x^5+58 x^4+76 x^3+29 x^2+39 x+28$
• $y^2=5 x^6+97 x^5+64 x^4+111 x^3+27 x^2+93 x+76$
• $y^2=45 x^6+103 x^5+86 x^4+61 x^3+108 x^2+16 x+7$
• $y^2=38 x^6+35 x^5+4 x^4+64 x^3+14 x^2+67 x+72$
• $y^2=30 x^6+25 x^5+94 x^4+37 x^3+73 x^2+92 x+86$
• $y^2=71 x^6+83 x^5+65 x^4+105 x^3+35 x^2+97 x+6$
• $y^2=59 x^6+29 x^5+22 x^4+7 x^3+89 x^2+25 x+16$
• $y^2=57 x^6+62 x^5+17 x^4+58 x^3+65 x^2+19 x+46$
• $y^2=85 x^6+85 x^5+98 x^4+104 x^3+71 x^2+39 x+97$
• $y^2=7 x^6+58 x^5+106 x^4+28 x^3+33 x+34$
• $y^2=96 x^6+14 x^5+86 x^4+12 x^3+59 x^2+63 x+27$
• $y^2=76 x^6+27 x^5+56 x^4+99 x^2+103 x+93$
• $y^2=73 x^6+102 x^5+16 x^4+43 x^3+4 x^2+92 x+68$
• $y^2=62 x^6+17 x^5+110 x^4+102 x^3+109 x^2+73 x+30$
• $y^2=9 x^6+55 x^5+3 x^4+95 x^3+97 x^2+5 x+41$
• $y^2=5 x^6+73 x^5+88 x^4+104 x^3+70 x^2+16 x+68$
• $y^2=37 x^6+92 x^5+17 x^4+96 x^3+58 x^2+21 x+110$
• $y^2=45 x^6+53 x^5+7 x^4+32 x^3+67 x^2+95 x+57$
• $y^2=101 x^6+50 x^5+24 x^4+102 x^3+3 x^2+110 x+2$
• $y^2=101 x^6+22 x^5+5 x^4+77 x^3+78 x^2+71 x+79$
• $y^2=73 x^6+5 x^5+19 x^4+7 x^3+10 x^2+36 x+72$
• $y^2=23 x^6+51 x^5+98 x^4+57 x^3+80 x^2+12 x+29$
• $y^2=27 x^6+61 x^5+37 x^4+73 x^3+92 x^2+88 x+43$
• $y^2=74 x^6+14 x^5+41 x^4+74 x^3+111 x^2+66 x+60$
• $y^2=10 x^6+110 x^5+56 x^4+110 x^3+70 x^2+94 x+108$
• $y^2=64 x^6+68 x^5+80 x^4+57 x^3+2 x^2+100 x+84$
• $y^2=77 x^6+111 x^5+12 x^4+45 x^3+50 x^2+62 x+92$
• $y^2=68 x^6+7 x^5+36 x^4+60 x^3+66 x^2+96 x+60$
• $y^2=49 x^6+89 x^5+50 x^4+84 x^3+42 x^2+6 x+83$
• $y^2=42 x^6+7 x^5+83 x^4+79 x^3+39 x^2+89 x+87$
• $y^2=47 x^6+49 x^5+56 x^4+28 x^3+51 x^2+23 x+48$
• $y^2=73 x^6+70 x^5+65 x^4+110 x^2+101 x+13$
• $y^2=6 x^6+107 x^5+32 x^4+32 x^3+24 x^2+11 x+74$
• $y^2=96 x^6+13 x^5+24 x^4+90 x^3+86 x^2+51 x+54$
• $y^2=66 x^6+93 x^5+9 x^4+59 x^3+55 x^2+73 x+101$
• $y^2=69 x^6+54 x^5+91 x^4+23 x^3+52 x^2+21 x+45$
• $y^2=80 x^6+20 x^5+59 x^4+93 x^3+2 x^2+92 x+84$
• $y^2=16 x^6+3 x^5+62 x^4+16 x^3+30 x^2+41 x+103$
• $y^2=13 x^6+19 x^5+67 x^4+89 x^3+54 x^2+20 x+1$
• $y^2=13 x^6+50 x^5+80 x^4+29 x^3+98 x^2+24 x+88$
• $y^2=74 x^6+111 x^5+16 x^4+104 x^3+62 x^2+22 x+70$
• $y^2=18 x^6+111 x^5+85 x^4+95 x^3+45 x^2+72 x+21$
• $y^2=105 x^6+28 x^5+69 x^4+14 x^3+28 x^2+8 x+26$
• $y^2=101 x^6+64 x^5+62 x^4+91 x^3+68 x^2+17 x+58$
• $y^2=27 x^6+21 x^5+91 x^4+22 x^3+35 x^2+12 x+93$
• $y^2=8 x^6+59 x^5+107 x^4+53 x^3+63 x^2+11 x+14$
• $y^2=36 x^6+71 x^5+85 x^4+72 x^3+97 x^2+112 x+69$
• $y^2=48 x^6+62 x^5+6 x^4+8 x^3+65 x^2+36 x+42$
• $y^2=59 x^6+32 x^5+38 x^4+96 x^3+108 x^2+19 x+89$
• $y^2=94 x^6+25 x^5+34 x^4+29 x^3+95 x^2+47 x+65$
• $y^2=23 x^6+61 x^5+66 x^4+103 x^3+71 x^2+45 x+25$
• $y^2=26 x^6+8 x^5+66 x^4+66 x^3+23 x^2+30 x+21$
• $y^2=43 x^6+21 x^5+85 x^4+63 x^3+32 x^2+87 x+96$
• $y^2=85 x^6+68 x^5+92 x^4+47 x^3+57 x^2+12 x+48$
• $y^2=95 x^6+88 x^5+99 x^4+61 x^3+83 x^2+85 x+27$
• $y^2=26 x^6+7 x^5+23 x^4+34 x^3+15 x^2+86 x+33$
• $y^2=92 x^6+88 x^5+75 x^4+18 x^3+85 x^2+16 x+11$
• $y^2=65 x^6+84 x^5+74 x^4+39 x^3+106 x^2+74 x+75$
• $y^2=73 x^6+27 x^5+16 x^4+47 x^3+85 x^2+79 x+94$
• $y^2=42 x^6+80 x^5+111 x^4+21 x^3+50 x^2+106 x+25$
• $y^2=107 x^6+66 x^5+22 x^4+36 x^3+15 x^2+59 x+72$
• $y^2=80 x^6+94 x^5+9 x^4+60 x^3+33 x^2+112 x+22$
• $y^2=58 x^6+98 x^5+32 x^4+14 x^3+62 x^2+100 x+68$
• $y^2=10 x^6+41 x^5+49 x^4+32 x^3+105 x^2+82 x+24$
• $y^2=61 x^6+67 x^5+96 x^4+88 x^3+59 x^2+112 x+23$
• $y^2=23 x^6+18 x^5+89 x^4+63 x^3+46 x^2+14 x+2$
• $y^2=77 x^6+71 x^5+13 x^4+13 x^3+47 x^2+20 x+78$
• $y^2=84 x^6+85 x^5+103 x^4+5 x^3+75 x^2+107 x+75$
• $y^2=71 x^6+111 x^5+34 x^4+61 x^3+14 x^2+92 x+110$
• $y^2=70 x^6+111 x^5+101 x^4+99 x^3+64 x^2+64 x+80$
• $y^2=86 x^6+42 x^5+32 x^4+68 x^3+31 x^2+16 x+9$
• $y^2=77 x^6+71 x^5+58 x^4+72 x^3+53 x^2+63 x+29$
• $y^2=107 x^6+39 x^5+104 x^4+100 x^3+82 x^2+9 x+53$
• $y^2=70 x^6+4 x^5+17 x^4+36 x^3+14 x^2+71 x+87$
• $y^2=78 x^6+81 x^5+34 x^4+54 x^3+22 x^2+72 x+40$
• $y^2=94 x^6+19 x^5+11 x^4+72 x^3+86 x^2+74 x+38$

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 4.0.381952109.1.

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.z_nz	$2$	(not in LMFDB)