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

• Label: 2.113.abe_qk
• 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:	$( 1 - 20 x + 113 x^{2} )( 1 - 10 x + 113 x^{2} )$
	$1 - 30 x + 426 x^{2} - 3390 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.110150159186$, $\pm0.344123913111$
Angle rank:	$2$ (numerical)
Jacobians:	$100$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9776$	$162438016$	$2083639913264$	$26585245054590976$	$339454337407338975536$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$12722$	$1444068$	$163052286$	$18424221444$	$2081950323698$	$235260562459188$	$26584442465450878$	$3004041944755770804$	$339456739033977209522$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 100 curves (of which all are hyperelliptic):

• $y^2=108 x^6+86 x^5+8 x^4+104 x^3+49 x^2+31 x+25$
• $y^2=10 x^6+105 x^5+63 x^4+89 x^3+87 x^2+10 x+108$
• $y^2=76 x^6+96 x^5+75 x^4+65 x^3+80 x^2+61 x+73$
• $y^2=25 x^6+3 x^5+23 x^4+35 x^3+46 x^2+62 x+8$
• $y^2=91 x^6+74 x^5+98 x^4+104 x^3+107 x^2+86 x+17$
• $y^2=45 x^6+62 x^5+78 x^4+8 x^3+14 x^2+41 x+12$
• $y^2=71 x^6+31 x^5+4 x^4+86 x^3+78 x^2+43 x+93$
• $y^2=92 x^6+65 x^5+4 x^4+33 x^3+36 x^2+97 x+75$
• $y^2=81 x^6+110 x^5+105 x^4+52 x^3+58 x^2+14 x+12$
• $y^2=58 x^6+66 x^5+89 x^4+26 x^3+66 x^2+33 x+5$
• $y^2=60 x^6+100 x^5+75 x^4+11 x^3+22 x+53$
• $y^2=55 x^6+43 x^5+109 x^4+24 x^3+43 x^2+54 x+112$
• $y^2=11 x^6+84 x^5+25 x^4+88 x^3+63 x^2+56 x+83$
• $y^2=84 x^6+110 x^5+99 x^4+67 x^3+69 x^2+74 x+85$
• $y^2=17 x^6+39 x^5+79 x^4+24 x^3+32 x^2+51 x+97$
• $y^2=58 x^6+108 x^5+83 x^4+90 x^3+100 x+40$
• $y^2=92 x^6+67 x^5+66 x^4+80 x^3+112 x^2+15 x+7$
• $y^2=17 x^6+18 x^5+11 x^4+104 x^3+38 x^2+83 x+107$
• $y^2=29 x^6+49 x^5+8 x^4+95 x^3+8 x^2+49 x+29$
• $y^2=69 x^6+60 x^5+19 x^4+27 x^3+104 x^2+47 x+12$
• and

• $y^2=102 x^6+9 x^5+69 x^4+54 x^3+69 x^2+9 x+102$
• $y^2=40 x^6+91 x^5+60 x^4+86 x^3+29 x^2+57 x+54$
• $y^2=33 x^6+46 x^5+72 x^4+86 x^3+80 x^2+33 x+42$
• $y^2=58 x^6+101 x^5+52 x^4+21 x^3+10 x^2+33 x+35$
• $y^2=45 x^6+68 x^5+24 x^4+77 x^3+112 x^2+107 x+104$
• $y^2=108 x^6+17 x^5+110 x^4+93 x^3+36 x^2+62 x+97$
• $y^2=81 x^6+87 x^5+10 x^4+53 x^3+91 x^2+64 x+64$
• $y^2=16 x^6+38 x^5+22 x^4+39 x^3+50 x^2+56 x+27$
• $y^2=16 x^6+14 x^5+66 x^4+91 x^3+3 x^2+64 x+18$
• $y^2=12 x^6+101 x^5+5 x^4+x^3+81 x^2+78 x+29$
• $y^2=111 x^6+2 x^5+46 x^4+7 x^3+52 x^2+88 x+110$
• $y^2=112 x^6+60 x^5+88 x^4+65 x^3+43 x^2+92 x+39$
• $y^2=79 x^6+89 x^5+19 x^4+86 x^3+19 x^2+89 x+79$
• $y^2=108 x^6+94 x^5+43 x^4+23 x^3+37 x^2+21 x+3$
• $y^2=2 x^6+52 x^5+95 x^4+68 x^3+45 x^2+33 x+112$
• $y^2=35 x^6+111 x^5+111 x^4+112 x^2+56 x+75$
• $y^2=29 x^6+77 x^5+93 x^4+2 x^3+34 x^2+15 x+84$
• $y^2=31 x^6+109 x^5+20 x^4+69 x^3+12 x^2+98 x+98$
• $y^2=20 x^6+x^5+14 x^4+82 x^3+80 x^2+14 x$
• $y^2=94 x^6+49 x^5+71 x^4+22 x^3+41 x^2+3 x+103$
• $y^2=79 x^6+61 x^5+54 x^4+91 x^3+108 x^2+5 x+9$
• $y^2=75 x^6+22 x^5+14 x^4+89 x^3+57 x^2+81 x+101$
• $y^2=68 x^6+32 x^5+30 x^4+63 x^3+25 x^2+13 x+108$
• $y^2=55 x^6+8 x^5+x^4+78 x^3+82 x^2+59 x+88$
• $y^2=76 x^6+29 x^5+111 x^4+51 x^3+100 x^2+15 x+31$
• $y^2=109 x^6+68 x^5+54 x^4+14 x^3+59 x^2+59 x+111$
• $y^2=39 x^6+57 x^5+59 x^4+64 x^3+49 x^2+67 x+76$
• $y^2=67 x^6+45 x^5+32 x^4+25 x^3+90 x^2+71 x+99$
• $y^2=22 x^6+33 x^5+17 x^4+9 x^3+64 x^2+92 x+84$
• $y^2=46 x^6+x^5+14 x^4+17 x^3+92 x^2+96 x+107$
• $y^2=17 x^6+64 x^5+24 x^4+14 x^3+39 x^2+93 x+101$
• $y^2=98 x^6+18 x^5+63 x^4+106 x^3+93 x^2+28 x+108$
• $y^2=83 x^6+35 x^5+53 x^4+5 x^3+11 x^2+41 x+71$
• $y^2=44 x^6+33 x^5+74 x^4+11 x^3+45 x^2+73 x+25$
• $y^2=7 x^6+111 x^5+96 x^4+25 x^3+19 x^2+72 x+29$
• $y^2=107 x^6+76 x^5+58 x^4+20 x^3+16 x^2+101 x+102$
• $y^2=43 x^6+60 x^5+10 x^4+91 x^3+32 x^2+53 x+9$
• $y^2=53 x^6+32 x^5+48 x^4+99 x^3+48 x^2+32 x+53$
• $y^2=80 x^6+88 x^5+23 x^4+98 x^3+23 x^2+88 x+80$
• $y^2=101 x^6+85 x^5+99 x^4+60 x^3+103 x^2+31 x+80$
• $y^2=12 x^6+77 x^5+6 x^4+28 x^3+4 x^2+110 x+34$
• $y^2=39 x^6+29 x^5+47 x^4+58 x^3+6 x^2+110 x+50$
• $y^2=54 x^6+89 x^5+9 x^4+10 x^3+111 x^2+16 x+87$
• $y^2=26 x^6+100 x^5+77 x^4+37 x^3+82 x^2+39 x+39$
• $y^2=63 x^6+45 x^5+88 x^4+93 x^3+46 x^2+50 x+87$
• $y^2=86 x^6+20 x^5+46 x^4+88 x^3+65 x^2+19 x+70$
• $y^2=59 x^6+101 x^5+106 x^4+42 x^3+95 x^2+50 x+35$
• $y^2=23 x^6+31 x^5+98 x^4+23 x^3+22 x^2+94 x+89$
• $y^2=84 x^6+72 x^5+27 x^4+70 x^3+108 x^2+96 x+73$
• $y^2=54 x^6+24 x^5+48 x^4+20 x^3+69 x^2+95 x+107$
• $y^2=27 x^6+53 x^5+45 x^4+91 x^3+59 x^2+36 x+24$
• $y^2=66 x^6+42 x^5+68 x^4+10 x^3+96 x^2+25 x+65$
• $y^2=17 x^6+79 x^5+5 x^4+102 x^3+42 x^2+46 x+3$
• $y^2=65 x^6+112 x^5+96 x^4+95 x^3+26 x^2+34 x+17$
• $y^2=92 x^6+47 x^5+61 x^4+73 x^3+17 x^2+97 x+84$
• $y^2=64 x^6+5 x^5+4 x^4+11 x^3+93 x^2+55 x+65$
• $y^2=66 x^6+97 x^5+77 x^3+64 x^2+72 x+47$
• $y^2=92 x^6+27 x^5+58 x^4+41 x^3+100 x^2+77 x+105$
• $y^2=50 x^6+105 x^5+88 x^4+96 x^3+65 x^2+101 x+26$
• $y^2=74 x^6+46 x^5+13 x^4+74 x^3+49 x^2+14 x+41$
• $y^2=79 x^6+98 x^5+55 x^4+38 x^3+55 x^2+98 x+79$
• $y^2=53 x^6+51 x^5+40 x^4+23 x^3+87 x^2+2 x+81$
• $y^2=93 x^6+48 x^5+60 x^4+6 x^3+4 x^2+60 x+70$
• $y^2=54 x^6+46 x^5+98 x^4+53 x^3+43 x^2+26 x+40$
• $y^2=22 x^6+12 x^5+80 x^4+89 x^3+101 x^2+24 x+16$
• $y^2=94 x^6+111 x^5+6 x^4+66 x^3+69 x^2+50 x+70$
• $y^2=23 x^6+65 x^5+56 x^4+19 x^3+69 x^2+19 x+23$
• $y^2=79 x^6+36 x^5+50 x^4+28 x^3+102 x^2+2 x$
• $y^2=70 x^6+24 x^5+26 x^4+77 x^3+39 x^2+7 x+15$
• $y^2=91 x^6+109 x^5+11 x^4+65 x^3+89 x^2+73 x+79$
• $y^2=66 x^6+99 x^5+85 x^4+40 x^3+14 x^2+78 x+75$
• $y^2=105 x^6+38 x^5+58 x^4+41 x^3+97 x^2+108 x+40$
• $y^2=43 x^6+87 x^5+x^4+13 x^3+65 x^2+103 x+1$
• $y^2=73 x^6+23 x^5+18 x^4+18 x^2+23 x+73$
• $y^2=79 x^6+2 x^5+34 x^4+30 x^3+35 x^2+16 x+59$
• $y^2=112 x^6+89 x^5+87 x^4+76 x^3+104 x^2+68 x+64$
• $y^2=64 x^6+35 x^5+19 x^4+77 x^3+64 x^2+61 x+40$
• $y^2=96 x^6+46 x^5+34 x^4+30 x^3+34 x^2+46 x+96$
• $y^2=96 x^5+107 x^4+99 x^3+20 x^2+14 x+43$
• $y^2=59 x^6+90 x^5+10 x^4+45 x^3+52 x^2+83 x+61$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{113}$

The isogeny class factors as 1.113.au $\times$ 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.au : \(\Q(\sqrt{-13}) \).
• 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.ak_ba	$2$	(not in LMFDB)
2.113.k_ba	$2$	(not in LMFDB)
2.113.be_qk	$2$	(not in LMFDB)