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

• Label: 2.113.abk_va
• 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:	$( 1 - 20 x + 113 x^{2} )( 1 - 16 x + 113 x^{2} )$
	$1 - 36 x + 546 x^{2} - 4068 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.110150159186$, $\pm0.228810695365$
Angle rank:	$2$ (numerical)
Jacobians:	$36$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9212$	$160473040$	$2082108851228$	$26587686780928000$	$339462031324100340572$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$78$	$12566$	$1443006$	$163067262$	$18424639038$	$2081954272214$	$235260561489486$	$26584441951774078$	$3004041938042961198$	$339456739004455819286$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 36 curves (of which all are hyperelliptic):

• $y^2=71 x^6+98 x^5+108 x^4+40 x^3+27 x^2+105 x+17$
• $y^2=70 x^6+58 x^5+20 x^4+78 x^3+68 x^2+110 x+89$
• $y^2=104 x^6+56 x^5+43 x^4+95 x^3+51 x^2+19 x+79$
• $y^2=27 x^6+105 x^5+84 x^4+63 x^3+73 x^2+7 x+7$
• $y^2=87 x^6+25 x^5+54 x^4+71 x^3+54 x^2+25 x+87$
• $y^2=42 x^6+52 x^5+85 x^4+101 x^3+17 x^2+71 x+16$
• $y^2=34 x^6+88 x^5+48 x^4+37 x^3+5 x^2+90 x+110$
• $y^2=17 x^6+90 x^5+76 x^4+73 x^3+11 x^2+33 x+52$
• $y^2=62 x^6+48 x^5+93 x^4+29 x^3+102 x^2+19 x+10$
• $y^2=35 x^6+103 x^5+106 x^4+41 x^3+77 x^2+33 x+84$
• $y^2=40 x^6+68 x^5+24 x^4+93 x^3+24 x^2+68 x+40$
• $y^2=97 x^6+66 x^5+82 x^4+106 x^3+16 x^2+x+5$
• $y^2=75 x^6+43 x^5+66 x^4+51 x^3+66 x^2+43 x+75$
• $y^2=77 x^6+21 x^5+9 x^4+107 x^3+41 x^2+47 x+96$
• $y^2=80 x^6+75 x^5+100 x^4+9 x^3+52 x^2+70 x+78$
• $y^2=84 x^6+61 x^5+9 x^4+19 x^3+17 x^2+85 x+66$
• $y^2=49 x^6+96 x^5+68 x^4+63 x^3+84 x^2+107 x+98$
• $y^2=17 x^6+15 x^5+54 x^4+8 x^3+54 x^2+15 x+17$
• $y^2=70 x^6+90 x^5+99 x^4+83 x^3+99 x^2+90 x+70$
• $y^2=88 x^6+110 x^5+66 x^4+81 x^3+9 x^2+28 x+3$
• and

• $y^2=103 x^6+107 x^5+96 x^4+59 x^3+46 x^2+46 x+54$
• $y^2=33 x^6+6 x^5+81 x^4+63 x^3+81 x^2+6 x+33$
• $y^2=59 x^6+107 x^5+42 x^4+25 x^3+47 x^2+57 x+64$
• $y^2=106 x^6+27 x^5+110 x^4+54 x^3+80 x^2+103 x+62$
• $y^2=95 x^6+85 x^5+61 x^4+51 x^3+11 x^2+112 x+49$
• $y^2=90 x^6+48 x^5+74 x^4+29 x^3+74 x^2+48 x+90$
• $y^2=35 x^6+76 x^5+74 x^4+10 x^3+76 x^2+19 x+38$
• $y^2=105 x^5+44 x^4+32 x^3+58 x^2+60 x+107$
• $y^2=47 x^6+49 x^5+91 x^4+5 x^3+91 x^2+49 x+47$
• $y^2=43 x^6+81 x^5+106 x^4+37 x^3+87 x^2+112 x+6$
• $y^2=92 x^6+94 x^5+50 x^4+70 x^3+82 x^2+33 x+31$
• $y^2=94 x^6+24 x^5+95 x^4+85 x^3+108 x^2+12 x+33$
• $y^2=21 x^6+15 x^5+68 x^4+23 x^3+79 x^2+108 x+92$
• $y^2=98 x^6+29 x^5+x^4+66 x^3+36 x^2+8 x+58$
• $y^2=x^6+57 x^5+24 x^4+109 x^3+24 x^2+57 x+1$
• $y^2=53 x^6+79 x^5+91 x^4+31 x^3+40 x^2+68 x+24$

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.aq 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.aq : \(\Q(\sqrt{-1}) \).

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.ae_adq	$2$	(not in LMFDB)
2.113.e_adq	$2$	(not in LMFDB)
2.113.bk_va	$2$	(not in LMFDB)