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

• Label: 2.113.abb_pg
• 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:	$( 1 - 17 x + 113 x^{2} )( 1 - 10 x + 113 x^{2} )$
	$1 - 27 x + 396 x^{2} - 3051 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.205038125192$, $\pm0.344123913111$
Angle rank:	$2$ (numerical)
Jacobians:	$24$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10088$	$163869472$	$2086631659424$	$26589534595721344$	$339458529331668337448$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$87$	$12833$	$1446138$	$163078593$	$18424448967$	$2081951089598$	$235260545577207$	$26584442022677761$	$3004041938225353434$	$339456738962598011393$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):

• $y^2=94 x^6+86 x^5+4 x^4+36 x^3+77 x^2+70 x+56$
• $y^2=20 x^6+72 x^5+104 x^4+31 x^3+110 x^2+53 x+74$
• $y^2=99 x^6+42 x^5+15 x^4+50 x^3+22 x^2+48 x+111$
• $y^2=99 x^5+5 x^4+80 x^3+84 x^2+7 x+99$
• $y^2=14 x^6+94 x^5+6 x^4+80 x^3+27 x^2+8 x+61$
• $y^2=90 x^6+107 x^5+15 x^4+9 x^3+39 x^2+20 x+105$
• $y^2=19 x^6+100 x^5+4 x^4+32 x^3+55 x^2+99 x+97$
• $y^2=84 x^6+102 x^5+100 x^4+91 x^3+85 x^2+32 x+12$
• $y^2=17 x^6+51 x^5+8 x^4+32 x^3+72 x^2+44 x$
• $y^2=57 x^6+28 x^5+39 x^4+91 x^3+5 x^2+63 x+23$
• $y^2=74 x^6+54 x^5+40 x^4+21 x^3+89 x^2+64 x+72$
• $y^2=86 x^6+42 x^5+99 x^4+86 x^3+x^2+67 x+86$
• $y^2=103 x^6+93 x^5+46 x^4+63 x^3+54 x^2+91 x+45$
• $y^2=8 x^6+85 x^5+46 x^4+61 x^3+111 x^2+84 x+52$
• $y^2=107 x^6+63 x^5+93 x^4+20 x^3+43 x^2+58 x+11$
• $y^2=86 x^6+72 x^5+33 x^4+90 x^3+29 x^2+51 x+84$
• $y^2=79 x^6+13 x^5+33 x^4+95 x^3+90 x^2+110 x+87$
• $y^2=79 x^6+103 x^5+13 x^4+41 x^3+58 x^2+106 x+84$
• $y^2=78 x^6+2 x^5+102 x^4+97 x^3+17 x^2+106 x+53$
• $y^2=65 x^6+5 x^5+109 x^4+84 x^3+100 x^2+20 x+38$
• $y^2=24 x^6+107 x^5+106 x^4+9 x^3+34 x^2+83 x+89$
• $y^2=44 x^6+94 x^5+48 x^4+69 x^3+20 x^2+61 x+3$
• $y^2=91 x^6+18 x^5+83 x^4+58 x^3+33 x^2+27 x+45$
• $y^2=48 x^6+6 x^5+85 x^4+107 x^3+76 x^2+16 x+54$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{113}$

The isogeny class factors as 1.113.ar $\times$ 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.ar : \(\Q(\sqrt{-163}) \).
• 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.ah_ce	$2$	(not in LMFDB)
2.113.h_ce	$2$	(not in LMFDB)
2.113.bb_pg	$2$	(not in LMFDB)