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

• Label: 2.113.aw_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:	$( 1 - 12 x + 113 x^{2} )( 1 - 10 x + 113 x^{2} )$
	$1 - 22 x + 346 x^{2} - 2486 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.309095034261$, $\pm0.344123913111$
Angle rank:	$2$ (numerical)
Jacobians:	$64$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10608$	$165739392$	$2088785138544$	$26589085373620224$	$339452831193910996848$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$92$	$12978$	$1447628$	$163075838$	$18424139692$	$2081946336498$	$235260512671612$	$26584442133932926$	$3004041943974382844$	$339456739034340842418$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 64 curves (of which all are hyperelliptic):

• $y^2=102 x^6+45 x^5+95 x^4+27 x^3+14 x^2+90 x+105$
• $y^2=58 x^6+72 x^5+14 x^4+11 x^3+14 x^2+72 x+58$
• $y^2=90 x^6+15 x^5+70 x^4+54 x^3+110 x^2+97 x+59$
• $y^2=60 x^6+38 x^5+26 x^4+94 x^3+26 x^2+38 x+60$
• $y^2=111 x^6+16 x^5+11 x^4+41 x^3+52 x^2+97 x+30$
• $y^2=76 x^6+11 x^5+77 x^4+45 x^3+77 x^2+11 x+76$
• $y^2=61 x^6+39 x^5+33 x^4+77 x^3+38 x^2+10 x+44$
• $y^2=29 x^6+26 x^5+2 x^4+26 x^3+2 x^2+26 x+29$
• $y^2=64 x^6+47 x^5+105 x^4+90 x^3+105 x^2+47 x+64$
• $y^2=15 x^6+19 x^5+63 x^4+48 x^3+63 x^2+19 x+15$
• $y^2=3 x^6+44 x^5+43 x^4+67 x^3+43 x^2+44 x+3$
• $y^2=44 x^6+55 x^5+40 x^4+67 x^3+3 x^2+84 x+109$
• $y^2=102 x^5+62 x^4+98 x^3+62 x^2+102 x$
• $y^2=24 x^6+6 x^5+96 x^4+44 x^3+27 x^2+21 x+66$
• $y^2=71 x^6+36 x^5+12 x^4+88 x^3+12 x^2+36 x+71$
• $y^2=80 x^6+25 x^5+3 x^4+70 x^3+3 x^2+25 x+80$
• $y^2=61 x^6+112 x^5+65 x^4+82 x^3+68 x^2+49 x+44$
• $y^2=65 x^6+63 x^5+53 x^4+64 x^3+63 x^2+72 x+35$
• $y^2=101 x^6+78 x^5+53 x^4+67 x^3+53 x^2+78 x+101$
• $y^2=47 x^6+24 x^5+65 x^4+60 x^3+65 x^2+24 x+47$
• and

• $y^2=5 x^6+93 x^5+41 x^4+6 x^3+41 x^2+93 x+5$
• $y^2=38 x^6+13 x^5+5 x^4+98 x^3+5 x^2+13 x+38$
• $y^2=31 x^6+14 x^5+99 x^4+27 x^3+32 x^2+57 x+9$
• $y^2=43 x^6+62 x^5+18 x^4+65 x^3+18 x^2+62 x+43$
• $y^2=107 x^6+91 x^5+87 x^4+76 x^3+87 x^2+91 x+107$
• $y^2=107 x^6+23 x^5+20 x^4+97 x^3+20 x^2+23 x+107$
• $y^2=46 x^6+105 x^5+63 x^4+18 x^3+99 x^2+x+108$
• $y^2=5 x^6+65 x^5+13 x^4+12 x^3+22 x^2+21 x+93$
• $y^2=3 x^6+16 x^5+20 x^4+17 x^3+20 x^2+16 x+3$
• $y^2=98 x^6+46 x^5+43 x^4+71 x^3+101 x^2+21 x+102$
• $y^2=68 x^6+37 x^5+88 x^4+6 x^3+88 x^2+37 x+68$
• $y^2=41 x^6+106 x^5+19 x^4+5 x^3+43 x^2+64 x+31$
• $y^2=88 x^6+59 x^5+61 x^4+26 x^3+98 x^2+78 x+1$
• $y^2=18 x^6+85 x^5+50 x^4+41 x^3+77 x^2+7 x+62$
• $y^2=23 x^6+38 x^5+97 x^4+83 x^3+50 x^2+78 x+103$
• $y^2=6 x^6+91 x^5+73 x^4+56 x^3+73 x^2+91 x+6$
• $y^2=44 x^6+93 x^5+55 x^4+22 x^3+107 x^2+19 x+104$
• $y^2=58 x^6+29 x^5+16 x^4+74 x^3+16 x^2+29 x+58$
• $y^2=76 x^6+4 x^5+105 x^4+43 x^3+105 x^2+4 x+76$
• $y^2=6 x^6+68 x^5+102 x^4+90 x^3+60 x^2+90 x+86$
• $y^2=15 x^6+10 x^5+43 x^4+45 x^3+43 x^2+10 x+15$
• $y^2=84 x^6+26 x^5+97 x^4+33 x^3+97 x^2+26 x+84$
• $y^2=33 x^6+91 x^5+27 x^3+13 x+40$
• $y^2=33 x^6+67 x^5+103 x^4+8 x^3+5 x^2+45 x+10$
• $y^2=26 x^6+44 x^5+5 x^4+102 x^3+5 x^2+44 x+26$
• $y^2=112 x^6+78 x^5+6 x^4+96 x^3+39 x^2+75 x+22$
• $y^2=100 x^6+67 x^5+x^4+99 x^3+x^2+67 x+100$
• $y^2=91 x^6+48 x^5+81 x^4+12 x^3+11 x^2+71 x+49$
• $y^2=58 x^6+37 x^5+96 x^4+102 x^3+96 x^2+37 x+58$
• $y^2=29 x^6+104 x^5+72 x^4+41 x^3+77 x^2+26 x+67$
• $y^2=x^6+20 x^5+65 x^4+81 x^3+65 x^2+20 x+1$
• $y^2=51 x^6+85 x^5+63 x^4+111 x^3+83 x^2+8 x+30$
• $y^2=77 x^6+69 x^5+8 x^4+83 x^3+22 x^2+91 x+7$
• $y^2=3 x^6+70 x^5+32 x^4+105 x^3+41 x^2+86 x+70$
• $y^2=85 x^6+108 x^5+111 x^4+35 x^3+111 x^2+108 x+85$
• $y^2=42 x^6+38 x^5+67 x^4+17 x^3+67 x^2+38 x+42$
• $y^2=104 x^6+74 x^5+100 x^4+73 x^3+100 x^2+74 x+104$
• $y^2=22 x^6+2 x^5+47 x^4+105 x^3+79 x^2+30 x+57$
• $y^2=4 x^6+66 x^5+2 x^4+99 x^3+15 x^2+40 x+49$
• $y^2=71 x^6+75 x^5+62 x^4+2 x^3+62 x^2+75 x+71$
• $y^2=47 x^6+45 x^5+109 x^4+17 x^3+32 x^2+55 x+5$
• $y^2=3 x^5+25 x^4+52 x^3+25 x^2+3 x$
• $y^2=60 x^6+51 x^5+20 x^4+26 x^3+94 x^2+61 x+98$
• $y^2=47 x^6+109 x^5+47 x^4+38 x^3+70 x^2+x+39$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{113}$

The isogeny class factors as 1.113.am $\times$ 1.113.ak 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.am : \(\Q(\sqrt{-77}) \).
• 1.113.ak : \(\Q(\sqrt{-22}) \).

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.ac_ec	$2$	(not in LMFDB)
2.113.c_ec	$2$	(not in LMFDB)
2.113.w_ni	$2$	(not in LMFDB)