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

• Label: 2.113.av_is
• Base field: $\F_{113}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• 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:	$( 1 - 21 x + 113 x^{2} )( 1 + 113 x^{2} )$
	$1 - 21 x + 226 x^{2} - 2373 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.0498602789898$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$104$

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10602$	$163164780$	$2078863950888$	$26576905919155200$	$339453194533848536922$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$93$	$12781$	$1440756$	$163001137$	$18424159413$	$2081952936034$	$235260534034101$	$26584441501112353$	$3004041937428477108$	$339456739028909272861$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 104 curves (of which all are hyperelliptic):

• $y^2=68 x^6+101 x^5+79 x^4+44 x^3+9 x^2+5 x+16$
• $y^2=39 x^6+39 x^5+75 x^4+35 x^3+79 x^2+31 x+73$
• $y^2=77 x^6+72 x^5+91 x^4+105 x^3+59 x^2+4 x+38$
• $y^2=93 x^6+41 x^5+110 x^4+40 x^3+102 x^2+41 x+55$
• $y^2=14 x^6+63 x^5+91 x^4+17 x^3+81 x^2+16 x+44$
• $y^2=47 x^6+85 x^5+88 x^4+56 x^3+57 x^2+4 x+29$
• $y^2=48 x^6+24 x^5+77 x^4+49 x^3+56 x^2+49 x+10$
• $y^2=15 x^6+60 x^5+86 x^4+2 x^3+41 x^2+41 x+86$
• $y^2=60 x^6+67 x^5+25 x^4+102 x^3+46 x^2+50 x+73$
• $y^2=73 x^6+4 x^5+62 x^4+100 x^3+71 x^2+46 x+21$
• $y^2=57 x^6+95 x^5+32 x^4+23 x^3+52 x^2+108 x+12$
• $y^2=30 x^6+65 x^5+91 x^4+111 x^3+53 x^2+69 x+72$
• $y^2=95 x^6+26 x^5+32 x^4+71 x^3+2 x^2+84 x+94$
• $y^2=63 x^6+56 x^5+63 x^4+27 x^3+46 x^2+83 x+91$
• $y^2=103 x^6+89 x^5+24 x^4+58 x^3+55 x^2+36 x+75$
• $y^2=55 x^6+39 x^5+x^4+5 x^3+66 x^2+6 x+95$
• $y^2=73 x^6+83 x^5+38 x^4+29 x^3+46 x^2+46 x+36$
• $y^2=71 x^6+7 x^5+17 x^4+49 x^3+78 x^2+66 x+45$
• $y^2=110 x^6+45 x^5+80 x^4+51 x^3+13 x^2+20 x+50$
• $y^2=107 x^6+82 x^5+76 x^4+71 x^3+25 x^2+55 x+89$
• and

• $y^2=10 x^6+5 x^5+14 x^4+109 x^3+42 x^2+60 x+38$
• $y^2=31 x^6+107 x^5+33 x^4+48 x^3+62 x^2+61 x+73$
• $y^2=66 x^6+62 x^5+5 x^4+68 x^3+41 x^2+57 x+46$
• $y^2=47 x^6+34 x^5+43 x^4+70 x^3+7 x^2+50 x+112$
• $y^2=x^6+101 x^5+79 x^4+43 x^3+42 x^2+50 x+49$
• $y^2=62 x^6+6 x^5+90 x^4+63 x^3+47 x^2+41 x+110$
• $y^2=47 x^6+64 x^5+108 x^4+111 x^3+78 x^2+100 x+11$
• $y^2=37 x^6+75 x^5+106 x^4+111 x^3+46 x^2+29 x+71$
• $y^2=4 x^6+86 x^5+25 x^4+25 x^3+76 x^2+103 x+6$
• $y^2=110 x^6+58 x^5+35 x^4+33 x^3+18 x^2+92 x+21$
• $y^2=19 x^6+69 x^5+3 x^4+97 x^3+27 x^2+90 x+43$
• $y^2=85 x^5+53 x^4+69 x^3+4 x^2+39 x+75$
• $y^2=34 x^6+38 x^5+102 x^4+22 x^3+55 x^2+86 x+2$
• $y^2=35 x^6+30 x^5+110 x^4+74 x^3+74 x^2+98 x+29$
• $y^2=79 x^6+111 x^5+80 x^4+107 x^3+93 x^2+112 x+86$
• $y^2=14 x^6+34 x^4+13 x^3+66 x^2+29 x+3$
• $y^2=64 x^6+4 x^5+90 x^4+5 x^3+15 x^2+2 x+5$
• $y^2=94 x^6+109 x^5+71 x^4+68 x^3+49 x^2+74 x+85$
• $y^2=25 x^6+7 x^5+56 x^4+29 x^3+77 x^2+45 x+110$
• $y^2=26 x^6+36 x^5+36 x^4+77 x^3+27 x^2+39 x+15$
• $y^2=74 x^6+77 x^5+45 x^4+49 x^3+58 x^2+112 x+18$
• $y^2=44 x^6+72 x^5+85 x^4+11 x^3+112 x^2+64 x+93$
• $y^2=22 x^6+41 x^5+21 x^4+88 x^3+94 x^2+62 x+108$
• $y^2=28 x^6+68 x^5+23 x^4+20 x^3+15 x^2+82 x+19$
• $y^2=30 x^6+13 x^5+109 x^4+26 x^3+70 x^2+31 x+40$
• $y^2=6 x^6+5 x^5+25 x^4+40 x^3+50 x^2+33 x+7$
• $y^2=12 x^6+4 x^5+98 x^4+93 x^3+46 x^2+89 x+100$
• $y^2=19 x^6+2 x^5+37 x^4+3 x^3+32 x^2+21 x+17$
• $y^2=70 x^6+78 x^5+103 x^4+49 x^3+108 x^2+90 x+71$
• $y^2=21 x^6+34 x^5+50 x^4+81 x^3+86 x^2+16 x+44$
• $y^2=34 x^6+4 x^5+83 x^4+25 x^3+76 x^2+69 x+31$
• $y^2=17 x^6+106 x^5+45 x^4+75 x^3+101 x^2+44 x+53$
• $y^2=36 x^6+82 x^5+40 x^4+72 x^3+14 x^2+46 x+107$
• $y^2=57 x^6+96 x^5+61 x^4+22 x^3+43 x^2+88 x+102$
• $y^2=94 x^6+112 x^5+48 x^4+13 x^3+96 x^2+48 x+18$
• $y^2=50 x^6+25 x^5+45 x^4+77 x^3+83 x^2+100 x+67$
• $y^2=109 x^6+91 x^5+78 x^4+74 x^3+56 x^2+101 x+28$
• $y^2=54 x^6+70 x^5+78 x^4+13 x^3+3 x^2+11 x+86$
• $y^2=83 x^6+97 x^5+60 x^4+10 x^3+36 x^2+40 x+40$
• $y^2=12 x^6+16 x^5+61 x^4+31 x^3+22 x^2+87 x+33$
• $y^2=108 x^6+35 x^5+33 x^4+90 x^3+17 x^2+63 x+12$
• $y^2=67 x^6+53 x^5+63 x^4+34 x^3+46 x^2+34 x+85$
• $y^2=89 x^6+91 x^5+89 x^4+65 x^3+x^2+83 x+61$
• $y^2=14 x^6+61 x^5+53 x^4+28 x^3+86 x^2+107 x+57$
• $y^2=60 x^6+12 x^5+71 x^4+50 x^3+39 x^2+97 x+85$
• $y^2=79 x^6+88 x^5+101 x^4+75 x^3+84 x^2+87 x+59$
• $y^2=38 x^6+35 x^5+92 x^4+33 x^3+108 x^2+66 x+42$
• $y^2=103 x^6+15 x^5+66 x^4+14 x^3+84 x^2+81 x+66$
• $y^2=41 x^6+109 x^5+21 x^4+38 x^3+101 x^2+41 x+68$
• $y^2=71 x^6+82 x^5+83 x^4+31 x^3+65 x^2+59 x+98$
• $y^2=111 x^6+107 x^5+27 x^4+80 x^3+50 x^2+62 x+87$
• $y^2=90 x^6+112 x^5+37 x^4+42 x^3+85 x^2+17 x+9$
• $y^2=70 x^6+4 x^5+12 x^4+34 x^3+101 x^2+70 x+47$
• $y^2=9 x^6+77 x^5+16 x^4+75 x^3+8 x^2+72 x+30$
• $y^2=89 x^6+62 x^5+20 x^4+34 x^3+41 x^2+8 x+31$
• $y^2=112 x^6+45 x^5+73 x^4+46 x^3+4 x^2+36 x+83$
• $y^2=34 x^6+43 x^5+13 x^4+38 x^3+13 x^2+26 x+65$
• $y^2=3 x^6+88 x^5+38 x^4+26 x^3+55 x^2+31 x+29$
• $y^2=4 x^6+64 x^5+46 x^4+27 x^3+75 x^2+78 x+8$
• $y^2=107 x^6+83 x^5+75 x^4+78 x^3+31 x^2+75 x+91$
• $y^2=82 x^6+58 x^5+2 x^4+91 x^3+31 x^2+85 x+67$
• $y^2=9 x^6+39 x^5+3 x^4+28 x^3+51 x^2+43 x+81$
• $y^2=85 x^6+40 x^5+106 x^4+73 x^3+55 x^2+86 x+57$
• $y^2=31 x^6+87 x^5+110 x^4+80 x^3+43 x^2+110 x+94$
• $y^2=91 x^6+95 x^5+58 x^4+31 x^3+66 x^2+58 x+83$
• $y^2=32 x^6+82 x^5+73 x^4+95 x^3+30 x^2+71 x+109$
• $y^2=105 x^6+42 x^5+101 x^4+10 x^3+73 x^2+61 x+75$
• $y^2=54 x^6+35 x^5+16 x^4+20 x^3+94 x^2+38 x+109$
• $y^2=101 x^6+6 x^5+11 x^4+30 x^3+66 x^2+101 x+111$
• $y^2=70 x^6+85 x^5+x^4+82 x^3+65 x^2+59 x+74$
• $y^2=43 x^6+3 x^5+52 x^4+25 x^3+57 x^2+53 x+47$
• $y^2=36 x^6+47 x^5+95 x^4+3 x^3+67 x^2+106 x+48$
• $y^2=70 x^6+9 x^5+13 x^4+67 x^3+42 x^2+87 x+42$
• $y^2=107 x^6+93 x^5+82 x^4+54 x^3+2 x^2+59 x+38$
• $y^2=107 x^6+62 x^5+37 x^4+18 x^3+107 x^2+60 x+77$
• $y^2=55 x^6+86 x^5+99 x^4+69 x^3+108 x^2+14 x+62$
• $y^2=107 x^6+102 x^5+60 x^4+109 x^3+59 x^2+75 x+110$
• $y^2=99 x^6+107 x^5+39 x^4+14 x^3+79 x^2+60 x+2$
• $y^2=34 x^6+9 x^5+76 x^4+71 x^3+51 x^2+81 x+19$
• $y^2=101 x^6+27 x^5+6 x^4+46 x^3+19 x^2+98 x+16$
• $y^2=17 x^6+63 x^5+65 x^4+52 x^3+3 x^2+28 x+85$
• $y^2=52 x^6+34 x^5+39 x^4+20 x^3+82 x^2+66 x+33$
• $y^2=48 x^6+68 x^5+30 x^4+49 x^3+43 x^2+28 x+7$
• $y^2=37 x^6+50 x^5+79 x^4+15 x^3+5 x^2+99 x+75$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{113^{2}}$.

Endomorphism algebra over $\F_{113}$

The isogeny class factors as 1.113.av $\times$ 1.113.a 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.a : \(\Q(\sqrt{-113}) \).

Endomorphism algebra over $\overline{\F}_{113}$

The base change of $A$ to $\F_{113^{2}}$ is 1.12769.aih $\times$ 1.12769.is. The endomorphism algebra for each factor is:

• 1.12769.aih : \(\Q(\sqrt{-11}) \).
• 1.12769.is : the quaternion algebra over \(\Q\) ramified at $113$ and $\infty$.

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.v_is	$2$	(not in LMFDB)