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

• Label: 2.113.ay_mj
• 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:	$( 1 - 19 x + 113 x^{2} )( 1 - 5 x + 113 x^{2} )$
	$1 - 24 x + 321 x^{2} - 2712 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.148111132014$, $\pm0.424431965615$
Angle rank:	$2$ (numerical)
Jacobians:	$68$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10355$	$163888585$	$2083616200640$	$26583211138882825$	$339455531376490620275$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$90$	$12836$	$1444050$	$163039812$	$18424286250$	$2081954884574$	$235260609066090$	$26584442306865028$	$3004041936794474610$	$339456738967890881636$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 68 curves (of which all are hyperelliptic):

• $y^2=76 x^6+96 x^5+62 x^4+34 x^3+38 x^2+38 x+70$
• $y^2=34 x^6+35 x^5+82 x^4+42 x^3+82 x^2+35 x+34$
• $y^2=34 x^6+56 x^5+41 x^4+60 x^3+41 x^2+56 x+34$
• $y^2=5 x^6+48 x^5+38 x^4+73 x^3+61 x^2+78 x+100$
• $y^2=36 x^6+9 x^5+66 x^4+107 x^3+43 x^2+110 x+19$
• $y^2=93 x^6+34 x^5+109 x^4+37 x^3+112 x^2+70 x+33$
• $y^2=40 x^6+89 x^5+9 x^4+7 x^3+110 x^2+23 x+90$
• $y^2=58 x^6+70 x^5+99 x^4+97 x^3+44 x^2+61 x+4$
• $y^2=6 x^6+15 x^5+32 x^4+49 x^3+14 x^2+37 x+28$
• $y^2=107 x^6+56 x^5+29 x^4+62 x^3+103 x^2+89 x+33$
• $y^2=85 x^6+98 x^5+65 x^4+36 x^3+11 x^2+65 x+35$
• $y^2=37 x^6+43 x^5+94 x^4+14 x^3+78 x^2+84 x+65$
• $y^2=103 x^6+50 x^5+13 x^4+85 x^3+103 x^2+81 x+90$
• $y^2=29 x^6+82 x^5+80 x^4+69 x^3+57 x^2+5 x+21$
• $y^2=12 x^6+49 x^5+112 x^4+64 x^3+x^2+85 x+112$
• $y^2=62 x^6+50 x^5+98 x^4+5 x^3+98 x^2+50 x+62$
• $y^2=x^6+4 x^5+100 x^4+19 x^3+7 x+82$
• $y^2=84 x^6+77 x^5+16 x^4+74 x^3+59 x^2+31 x+83$
• $y^2=70 x^6+93 x^5+51 x^4+99 x^3+44 x^2+53 x+110$
• $y^2=67 x^6+86 x^5+22 x^4+18 x^3+18 x^2+110 x+60$
• and

• $y^2=94 x^6+19 x^5+11 x^4+88 x^3+47 x^2+47 x+80$
• $y^2=38 x^6+24 x^5+29 x^4+59 x^3+29 x^2+24 x+38$
• $y^2=23 x^6+81 x^5+97 x^4+19 x^3+73 x^2+x+62$
• $y^2=52 x^6+96 x^5+43 x^4+57 x^3+24 x^2+94 x+19$
• $y^2=98 x^6+72 x^5+12 x^3+111 x^2+58 x+55$
• $y^2=96 x^6+103 x^5+64 x^4+49 x^3+14 x^2+43 x+65$
• $y^2=71 x^6+4 x^5+58 x^4+70 x^3+18 x^2+25 x+75$
• $y^2=105 x^6+106 x^5+45 x^4+31 x^3+27 x^2+14 x+57$
• $y^2=45 x^6+44 x^5+54 x^4+25 x^3+65 x^2+80 x+107$
• $y^2=66 x^6+43 x^5+43 x^4+82 x^3+43 x^2+43 x+66$
• $y^2=63 x^6+36 x^5+64 x^4+60 x^3+81 x^2+84 x+8$
• $y^2=2 x^6+21 x^5+74 x^4+70 x^3+36 x^2+25 x+106$
• $y^2=69 x^6+47 x^5+15 x^4+108 x^3+15 x^2+47 x+69$
• $y^2=90 x^6+89 x^5+6 x^4+85 x^3+50 x^2+105 x+111$
• $y^2=90 x^6+100 x^5+82 x^4+66 x^3+110 x^2+31 x+101$
• $y^2=58 x^6+76 x^5+59 x^4+58 x^3+59 x^2+76 x+58$
• $y^2=22 x^6+108 x^5+7 x^4+85 x^3+48 x^2+32 x+109$
• $y^2=32 x^6+84 x^5+102 x^4+83 x^3+39 x^2+92 x+10$
• $y^2=39 x^6+24 x^5+88 x^4+78 x^3+67 x^2+96 x+103$
• $y^2=27 x^6+28 x^5+41 x^4+2 x^3+41 x^2+28 x+27$
• $y^2=48 x^6+48 x^5+20 x^4+90 x^3+82 x^2+45 x+53$
• $y^2=84 x^6+36 x^5+101 x^4+100 x^3+36 x^2+99 x+108$
• $y^2=31 x^6+69 x^5+50 x^4+103 x^3+50 x^2+69 x+31$
• $y^2=59 x^6+90 x^5+2 x^4+92 x^3+64 x^2+76 x+98$
• $y^2=70 x^6+44 x^5+103 x^4+6 x^3+55 x^2+104 x+73$
• $y^2=110 x^6+67 x^5+96 x^4+102 x^3+73 x^2+38 x+108$
• $y^2=98 x^6+62 x^5+6 x^4+31 x^3+12 x^2+34 x+90$
• $y^2=37 x^6+27 x^5+57 x^4+54 x^3+100 x^2+24 x+22$
• $y^2=32 x^6+73 x^5+66 x^4+10 x^3+91 x^2+15 x+13$
• $y^2=102 x^6+106 x^5+77 x^4+112 x^3+46 x^2+34 x+33$
• $y^2=88 x^6+5 x^5+29 x^4+75 x^3+4 x^2+80 x+108$
• $y^2=26 x^6+x^5+9 x^4+81 x^3+9 x^2+x+26$
• $y^2=93 x^6+109 x^5+99 x^4+34 x^3+8 x^2+50 x+43$
• $y^2=43 x^6+81 x^5+19 x^4+74 x^3+26 x^2+91 x+3$
• $y^2=79 x^6+14 x^5+66 x^4+21 x^3+101 x^2+57 x+60$
• $y^2=68 x^6+104 x^5+79 x^4+8 x^3+38 x^2+110 x+85$
• $y^2=85 x^6+76 x^5+88 x^4+80 x^3+97 x^2+53 x+72$
• $y^2=77 x^6+77 x^5+91 x^4+78 x^3+19 x^2+70 x+43$
• $y^2=108 x^6+31 x^5+69 x^4+93 x^3+x^2+19$
• $y^2=54 x^6+18 x^5+97 x^4+59 x^3+57 x^2+4 x+93$
• $y^2=12 x^6+64 x^5+93 x^4+108 x^3+107 x^2+13 x+42$
• $y^2=66 x^6+6 x^5+47 x^4+9 x^3+24 x^2+48 x+108$
• $y^2=108 x^6+35 x^5+87 x^4+34 x^3+101 x^2+11 x+17$
• $y^2=73 x^6+16 x^5+72 x^4+74 x^3+72 x^2+16 x+73$
• $y^2=12 x^6+66 x^5+58 x^4+35 x^3+58 x^2+66 x+12$
• $y^2=87 x^6+48 x^5+25 x^4+80 x^3+12 x^2+11 x+82$
• $y^2=39 x^6+100 x^5+95 x^4+109 x^3+11 x^2+45 x+73$
• $y^2=84 x^6+50 x^5+41 x^4+4 x^3+86 x^2+109 x+58$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{113}$

The isogeny class factors as 1.113.at $\times$ 1.113.af 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.at : \(\Q(\sqrt{-91}) \).
• 1.113.af : \(\Q(\sqrt{-427}) \).

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.ao_fb	$2$	(not in LMFDB)
2.113.o_fb	$2$	(not in LMFDB)
2.113.y_mj	$2$	(not in LMFDB)