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

• Label: 2.113.abi_tf
• 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:	$( 1 - 21 x + 113 x^{2} )( 1 - 13 x + 113 x^{2} )$
	$1 - 34 x + 499 x^{2} - 3842 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.0498602789898$, $\pm0.290579079721$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9393$	$161042985$	$2082048021648$	$26584702995940425$	$339453931997677361073$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$80$	$12612$	$1442966$	$163048964$	$18424199440$	$2081948051934$	$235260503533072$	$26584441656502276$	$3004041938655930998$	$339456739027307112132$

Jacobians and polarizations

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

• $y^2=67 x^6+69 x^5+51 x^4+13 x^3+44 x^2+44 x+70$
• $y^2=38 x^6+12 x^5+22 x^4+55 x^3+52 x^2+110 x+70$
• $y^2=3 x^6+30 x^5+84 x^4+87 x^3+3 x^2+78 x+55$
• $y^2=4 x^6+19 x^5+72 x^4+12 x^3+47 x^2+33 x+90$
• $y^2=82 x^6+60 x^5+19 x^4+8 x^3+28 x^2+74 x+2$
• $y^2=107 x^6+79 x^5+80 x^4+57 x^3+47 x^2+90 x+65$
• $y^2=54 x^6+78 x^5+67 x^4+38 x^3+3 x^2+83 x+72$
• $y^2=20 x^6+54 x^5+77 x^4+70 x^3+89 x^2+103 x+87$
• $y^2=63 x^6+18 x^5+104 x^4+54 x^3+72 x^2+64 x+64$
• $y^2=73 x^6+16 x^5+83 x^4+12 x^3+59 x^2+109 x+96$
• $y^2=53 x^6+19 x^5+16 x^4+92 x^3+52 x^2+69 x+101$
• $y^2=80 x^6+8 x^5+96 x^4+18 x^3+6 x^2+77 x+21$
• $y^2=53 x^6+79 x^5+57 x^4+93 x^3+13 x^2+25 x+67$
• $y^2=30 x^6+54 x^5+91 x^4+17 x^3+27 x^2+11 x+34$
• $y^2=47 x^6+45 x^5+95 x^4+86 x^3+44 x^2+63 x+104$
• $y^2=86 x^6+18 x^5+50 x^4+78 x^3+74 x^2+102 x+9$
• $y^2=84 x^6+97 x^5+100 x^4+59 x^3+22 x^2+103 x+43$
• $y^2=71 x^6+10 x^5+98 x^4+4 x^3+72 x^2+27 x+20$
• $y^2=19 x^6+87 x^5+58 x^4+75 x^3+70 x^2+41 x+65$
• $y^2=109 x^6+5 x^5+106 x^4+92 x^3+51 x^2+74 x+102$
• and

• $y^2=79 x^6+35 x^5+70 x^4+30 x^3+3 x^2+38 x+70$
• $y^2=13 x^6+102 x^5+79 x^4+14 x^3+57 x^2+28 x+21$
• $y^2=46 x^6+60 x^4+99 x^3+26 x^2+75$
• $y^2=94 x^6+45 x^5+99 x^4+70 x^3+53 x^2+29 x+28$
• $y^2=29 x^6+76 x^5+111 x^4+2 x^3+87 x^2+75 x+94$
• $y^2=99 x^6+15 x^5+21 x^4+28 x^3+67 x^2+74 x+12$
• $y^2=42 x^6+73 x^5+83 x^4+21 x^3+76 x^2+39 x+64$
• $y^2=60 x^6+99 x^5+5 x^4+24 x^3+51 x^2+38 x+97$
• $y^2=75 x^6+75 x^5+87 x^4+9 x^3+87 x^2+72 x+101$
• $y^2=33 x^6+32 x^5+12 x^4+87 x^3+37 x^2+16 x+103$
• $y^2=91 x^6+43 x^5+17 x^4+34 x^3+8 x^2+59 x+3$
• $y^2=71 x^6+21 x^5+15 x^4+95 x^3+33 x^2+109 x+34$
• $y^2=45 x^6+91 x^5+86 x^4+18 x^3+109 x^2+42 x+33$
• $y^2=90 x^6+62 x^5+3 x^4+71 x^3+18 x^2+87 x+34$
• $y^2=80 x^6+112 x^5+49 x^4+72 x^3+28 x^2+85 x+37$
• $y^2=98 x^6+81 x^5+97 x^4+28 x^3+54 x^2+38 x+35$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{113}$

The isogeny class factors as 1.113.av $\times$ 1.113.an 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.an : \(\Q(\sqrt{-283}) \).

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.ai_abv	$2$	(not in LMFDB)
2.113.i_abv	$2$	(not in LMFDB)
2.113.bi_tf	$2$	(not in LMFDB)