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

• Label: 2.113.abk_ve
• 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:	$( 1 - 18 x + 113 x^{2} )^{2}$
	$1 - 36 x + 550 x^{2} - 4068 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.178616545187$, $\pm0.178616545187$
Angle rank:	$1$ (numerical)
Jacobians:	$46$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9216$	$160579584$	$2082733876224$	$26589638502383616$	$339466183491188745216$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$78$	$12574$	$1443438$	$163079230$	$18424864398$	$2081957378398$	$235260591384750$	$26584442073469054$	$3004041935686141134$	$339456738934538291614$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 46 curves (of which all are hyperelliptic):

• $y^2=47 x^6+93 x^5+42 x^4+61 x^3+12 x^2+86 x+94$
• $y^2=59 x^6+68 x^4+68 x^2+59$
• $y^2=21 x^6+110 x^5+31 x^4+45 x^3+31 x^2+110 x+21$
• $y^2=61 x^6+89 x^5+89 x^4+112 x^3+89 x^2+89 x+61$
• $y^2=20 x^6+6 x^5+92 x^4+84 x^3+78 x^2+92 x+34$
• $y^2=10 x^6+102 x^5+10 x^4+45 x^3+103 x^2+102 x+103$
• $y^2=47 x^6+15 x^4+15 x^2+47$
• $y^2=92 x^6+78 x^5+10 x^4+36 x^3+10 x^2+78 x+92$
• $y^2=67 x^6+34 x^5+62 x^4+100 x^3+62 x^2+34 x+67$
• $y^2=32 x^6+81 x^5+109 x^4+20 x^3+109 x^2+81 x+32$
• $y^2=8 x^6+57 x^5+87 x^4+22 x^3+87 x^2+57 x+8$
• $y^2=108 x^6+33 x^5+57 x^4+53 x^3+57 x^2+33 x+108$
• $y^2=52 x^6+90 x^5+15 x^4+88 x^3+15 x^2+90 x+52$
• $y^2=103 x^6+69 x^5+54 x^4+47 x^3+34 x^2+52 x+55$
• $y^2=102 x^6+71 x^5+94 x^4+92 x^3+94 x^2+71 x+102$
• $y^2=108 x^6+31 x^5+58 x^4+85 x^3+58 x^2+31 x+108$
• $y^2=30 x^6+100 x^4+100 x^2+30$
• $y^2=7 x^6+101 x^5+56 x^4+37 x^3+55 x^2+2 x+75$
• $y^2=43 x^6+22 x^5+94 x^4+94 x^3+101 x^2+24 x$
• $y^2=97 x^6+78 x^5+21 x^4+88 x^3+107 x^2+94 x+32$
• and

• $y^2=40 x^6+35 x^5+100 x^4+7 x^3+81 x^2+75 x+21$
• $y^2=46 x^6+20 x^4+20 x^2+46$
• $y^2=107 x^6+65 x^5+40 x^4+63 x^3+107 x^2+17 x+47$
• $y^2=x^5+112 x$
• $y^2=71 x^6+87 x^5+22 x^4+108 x^3+9 x^2+26 x+48$
• $y^2=101 x^6+76 x^4+59 x^3+76 x^2+101$
• $y^2=55 x^6+100 x^5+63 x^4+69 x^3+63 x^2+100 x+55$
• $y^2=30 x^6+79 x^5+74 x^4+21 x^3+58 x^2+17 x+31$
• $y^2=29 x^6+49 x^5+93 x^4+29 x^3+15 x^2+3 x+76$
• $y^2=19 x^6+10 x^5+111 x^4+39 x^3+111 x^2+10 x+19$
• $y^2=23 x^6+103 x^5+34 x^4+60 x^3+33 x^2+94 x+70$
• $y^2=48 x^6+64 x^5+86 x^4+15 x^3+94 x^2+83 x+3$
• $y^2=61 x^6+7 x^5+66 x^4+41 x^3+66 x^2+7 x+61$
• $y^2=66 x^6+48 x^5+103 x^4+19 x^3+33 x^2+21 x+5$
• $y^2=91 x^6+78 x^5+46 x^4+48 x^3+46 x^2+78 x+91$
• $y^2=112 x^6+67 x^5+62 x^4+40 x^3+13 x^2+50 x+71$
• $y^2=98 x^6+83 x^4+83 x^2+98$
• $y^2=75 x^6+92 x^5+95 x^4+73 x^3+3 x^2+102 x+48$
• $y^2=78 x^6+37 x^5+77 x^4+48 x^3+x^2+54 x+6$
• $y^2=109 x^6+8 x^5+74 x^4+56 x^3+84 x^2+112 x+25$
• $y^2=74 x^6+106 x^4+106 x^2+74$
• $y^2=96 x^6+47 x^5+9 x^4+52 x^3+104 x^2+47 x+17$
• $y^2=66 x^6+62 x^5+98 x^4+2 x^3+98 x^2+62 x+66$
• $y^2=20 x^6+63 x^5+112 x^4+70 x^3+60 x^2+22 x+10$
• $y^2=52 x^6+75 x^5+27 x^4+107 x^3+74 x^2+10 x+31$
• $y^2=103 x^6+4 x^5+11 x^4+108 x^3+104 x^2+97 x+70$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{113}$

The isogeny class factors as 1.113.as^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.113.a_adu	$2$	(not in LMFDB)
2.113.bk_ve	$2$	(not in LMFDB)
2.113.s_id	$3$	(not in LMFDB)