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

• Label: 2.113.abb_pm
• 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:	$( 1 - 16 x + 113 x^{2} )( 1 - 11 x + 113 x^{2} )$
	$1 - 27 x + 402 x^{2} - 3051 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.228810695365$, $\pm0.326901256467$
Angle rank:	$2$ (numerical)
Jacobians:	$27$
Isomorphism classes:	177

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10094$	$164027500$	$2087334060896$	$26590826080000000$	$339459185995325039294$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$87$	$12845$	$1446624$	$163086513$	$18424484607$	$2081950010210$	$235260519481599$	$26584441763595553$	$3004041937987073952$	$339456738993918837725$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 27 curves (of which all are hyperelliptic):

• $y^2=94 x^6+4 x^5+48 x^4+85 x^3+43 x^2+72 x+71$
• $y^2=5 x^6+54 x^5+100 x^4+109 x^3+49 x^2+70 x+100$
• $y^2=35 x^6+87 x^5+96 x^4+94 x^3+72 x^2+x+16$
• $y^2=88 x^6+106 x^5+30 x^4+5 x^3+68 x^2+54 x+68$
• $y^2=33 x^6+101 x^4+14 x^3+24 x^2+7 x+56$
• $y^2=9 x^6+3 x^5+31 x^4+89 x^3+36 x^2+84 x+54$
• $y^2=46 x^6+66 x^4+23 x^2+47 x+112$
• $y^2=27 x^6+102 x^5+77 x^4+110 x^3+36 x^2+61 x+84$
• $y^2=107 x^6+30 x^5+67 x^4+32 x^3+81 x^2+95 x+26$
• $y^2=97 x^6+12 x^5+104 x^4+108 x^3+58 x^2+21 x+73$
• $y^2=110 x^6+98 x^5+77 x^4+21 x^3+9 x^2+68 x+83$
• $y^2=26 x^6+28 x^5+58 x^4+65 x^3+59 x^2+88 x+34$
• $y^2=97 x^6+32 x^5+84 x^4+6 x^3+3 x^2+85 x+49$
• $y^2=31 x^6+34 x^5+77 x^4+19 x^3+45 x^2+91 x+15$
• $y^2=96 x^6+108 x^5+38 x^4+51 x^3+51 x^2+7 x+92$
• $y^2=56 x^6+60 x^5+65 x^4+83 x^3+29 x^2+14 x+38$
• $y^2=35 x^6+98 x^5+28 x^4+101 x^3+46 x^2+32 x+60$
• $y^2=27 x^6+4 x^5+13 x^4+96 x^3+82 x^2+71 x+34$
• $y^2=28 x^6+76 x^5+84 x^4+10 x^3+14 x^2+24 x+23$
• $y^2=37 x^6+44 x^5+46 x^4+69 x^3+85 x^2+92 x+111$
• $y^2=96 x^6+101 x^5+101 x^4+95 x^3+52 x^2+19 x+50$
• $y^2=96 x^6+76 x^4+104 x^3+58 x^2+2 x+71$
• $y^2=79 x^6+6 x^5+23 x^4+80 x^3+49 x^2+x+70$
• $y^2=15 x^6+61 x^5+9 x^4+72 x^3+65 x^2+83 x+5$
• $y^2=59 x^6+35 x^5+34 x^4+108 x^3+29 x^2+57 x+74$
• $y^2=59 x^6+22 x^5+13 x^4+17 x^3+75 x^2+33 x+13$
• $y^2=75 x^6+55 x^5+104 x^4+34 x^3+29 x^2+65 x+54$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{113}$

The isogeny class factors as 1.113.aq $\times$ 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.aq : \(\Q(\sqrt{-1}) \).
• 1.113.al : \(\Q(\sqrt{-331}) \).

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.af_by	$2$	(not in LMFDB)
2.113.f_by	$2$	(not in LMFDB)
2.113.bb_pm	$2$	(not in LMFDB)