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

• Label: 2.113.abg_rp
• 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:	$( 1 - 21 x + 113 x^{2} )( 1 - 11 x + 113 x^{2} )$
	$1 - 32 x + 457 x^{2} - 3616 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.0498602789898$, $\pm0.326901256467$
Angle rank:	$2$ (numerical)
Jacobians:	$33$
Isomorphism classes:	156

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9579$	$161645625$	$2082318883776$	$26583435300515625$	$339451143353756928219$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$82$	$12660$	$1443154$	$163041188$	$18424048082$	$2081947185630$	$235260515103794$	$26584441942674628$	$3004041940837744882$	$339456739016514681300$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 33 curves (of which all are hyperelliptic):

• $y^2=105 x^6+16 x^5+16 x^4+46 x^3+70 x^2+18 x+81$
• $y^2=37 x^6+x^5+6 x^4+31 x^3+13 x^2+108 x+19$
• $y^2=93 x^6+90 x^5+94 x^4+75 x^3+94 x^2+90 x+93$
• $y^2=48 x^6+22 x^5+76 x^4+31 x^3+2 x^2+101 x+92$
• $y^2=107 x^6+33 x^5+64 x^4+86 x^3+64 x^2+33 x+107$
• $y^2=48 x^6+47 x^5+39 x^4+22 x^3+21 x^2+72 x+68$
• $y^2=40 x^6+68 x^5+48 x^4+81 x^3+48 x^2+68 x+40$
• $y^2=111 x^6+21 x^5+10 x^4+35 x^3+54 x^2+31 x+65$
• $y^2=12 x^6+79 x^5+64 x^4+9 x^3+80 x^2+36 x+32$
• $y^2=108 x^6+74 x^5+107 x^4+37 x^3+99 x^2+69 x+53$
• $y^2=76 x^6+96 x^5+16 x^4+73 x^3+107 x^2+44 x+100$
• $y^2=9 x^6+84 x^5+71 x^4+22 x^3+84 x^2+72 x+8$
• $y^2=90 x^6+35 x^4+52 x^3+15 x^2+86 x+41$
• $y^2=92 x^6+44 x^5+23 x^4+71 x^3+53 x^2+27 x+14$
• $y^2=27 x^6+54 x^5+30 x^4+76 x^3+60 x^2+24 x+80$
• $y^2=65 x^6+76 x^5+7 x^4+72 x^3+6 x^2+53 x+88$
• $y^2=39 x^6+5 x^5+68 x^4+4 x^3+85 x^2+46 x+35$
• $y^2=2 x^6+45 x^5+102 x^4+81 x^3+62 x^2+79 x+36$
• $y^2=24 x^6+38 x^5+100 x^4+18 x^3+100 x^2+38 x+24$
• $y^2=107 x^6+35 x^5+36 x^4+27 x^3+36 x^2+35 x+107$
• and

• $y^2=16 x^6+74 x^5+13 x^4+100 x^3+48 x^2+16 x+55$
• $y^2=54 x^6+17 x^5+41 x^4+38 x^3+41 x^2+17 x+54$
• $y^2=90 x^6+88 x^5+60 x^4+57 x^3+56 x^2+12 x+48$
• $y^2=72 x^6+33 x^5+93 x^4+51 x^3+93 x^2+33 x+72$
• $y^2=56 x^6+56 x^5+18 x^4+51 x^3+39 x^2+11 x+112$
• $y^2=70 x^6+56 x^5+109 x^4+24 x^3+109 x^2+56 x+70$
• $y^2=90 x^6+111 x^5+33 x^4+65 x^3+43 x^2+50 x+18$
• $y^2=52 x^6+47 x^5+85 x^4+49 x^3+96 x^2+76 x+73$
• $y^2=37 x^6+74 x^5+82 x^4+85 x^3+40 x^2+39 x+73$
• $y^2=51 x^6+53 x^5+103 x^4+20 x^3+41 x^2+89 x+42$
• $y^2=37 x^6+51 x^5+4 x^4+92 x^3+66 x^2+12 x+67$
• $y^2=73 x^6+69 x^5+82 x^4+42 x^3+109 x^2+41 x+45$
• $y^2=103 x^6+13 x^5+109 x^4+107 x^3+109 x^2+13 x+103$

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.al 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.al : \(\Q(\sqrt{-331}) \).

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_af	$2$	(not in LMFDB)
2.113.k_af	$2$	(not in LMFDB)
2.113.bg_rp	$2$	(not in LMFDB)