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

• Label: 2.113.au_ie
• Base field: $\F_{113}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{113}$
Dimension:	$2$
L-polynomial:	$1 - 20 x + 212 x^{2} - 2260 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.0747295012116$, $\pm0.510138955442$
Angle rank:	$2$ (numerical)
Number field:	4.0.2296445184.1
Galois group:	$D_{4}$
Jacobians:	$64$
Isomorphism classes:	128

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10702$	$163333924$	$2078982688174$	$26577853567182736$	$339455598557176453582$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$94$	$12794$	$1440838$	$163006950$	$18424289894$	$2081954119898$	$235260543486398$	$26584441712157694$	$3004041940752992734$	$339456739053051079514$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 64 curves (of which all are hyperelliptic):

• $y^2=3 x^6+9 x^5+49 x^4+75 x^3+20 x^2+105 x+63$
• $y^2=43 x^6+89 x^5+65 x^4+25 x^3+17 x^2+50 x+102$
• $y^2=13 x^6+40 x^5+110 x^4+9 x^3+41 x^2+83 x+112$
• $y^2=109 x^6+102 x^5+112 x^4+57 x^3+6 x^2+26 x+109$
• $y^2=48 x^6+63 x^5+8 x^4+54 x^3+34 x^2+92 x+86$
• $y^2=80 x^6+83 x^5+27 x^4+43 x^3+49 x^2+5 x+107$
• $y^2=54 x^6+89 x^5+51 x^4+103 x^3+24 x^2+88 x+93$
• $y^2=7 x^6+3 x^5+21 x^4+31 x^3+89 x^2+54 x+80$
• $y^2=29 x^6+27 x^5+73 x^4+91 x^3+49 x^2+50 x+112$
• $y^2=88 x^6+22 x^5+93 x^4+69 x^3+55 x^2+58 x+59$
• $y^2=61 x^6+92 x^5+68 x^4+103 x^3+100 x^2+54 x+33$
• $y^2=22 x^6+44 x^5+50 x^4+34 x^3+37 x^2+37 x+18$
• $y^2=7 x^6+43 x^5+41 x^4+24 x^3+38 x^2+61 x+93$
• $y^2=85 x^6+3 x^5+61 x^4+29 x^3+29 x^2+8 x+47$
• $y^2=58 x^6+74 x^5+97 x^4+101 x^3+112 x^2+86 x+42$
• $y^2=65 x^6+63 x^5+90 x^4+17 x^3+112 x^2+18 x+51$
• $y^2=89 x^6+29 x^5+36 x^4+19 x^3+x^2+10 x+48$
• $y^2=103 x^6+62 x^5+26 x^4+93 x^3+45 x^2+80 x+45$
• $y^2=45 x^6+23 x^5+33 x^4+42 x^3+89 x^2+99 x+49$
• $y^2=16 x^6+21 x^5+54 x^4+64 x^3+105 x^2+58 x+21$
• and

• $y^2=107 x^6+70 x^5+26 x^4+11 x^3+37 x^2+57 x+60$
• $y^2=35 x^6+101 x^5+83 x^4+7 x^3+53 x^2+91 x+90$
• $y^2=96 x^6+39 x^5+92 x^4+18 x^3+38 x^2+58 x+38$
• $y^2=87 x^6+71 x^5+17 x^4+19 x^3+93 x^2+36 x+33$
• $y^2=44 x^6+34 x^5+88 x^4+98 x^3+43 x^2+82 x+49$
• $y^2=74 x^6+11 x^5+29 x^4+91 x^3+58 x^2+23 x+44$
• $y^2=62 x^6+108 x^5+56 x^4+99 x^3+87 x^2+69 x+34$
• $y^2=41 x^6+9 x^5+27 x^4+88 x^3+88 x^2+104 x+69$
• $y^2=107 x^6+28 x^5+82 x^4+71 x^3+25 x^2+88 x+33$
• $y^2=88 x^6+99 x^5+57 x^4+3 x^3+73 x^2+28 x+83$
• $y^2=84 x^6+35 x^5+92 x^4+83 x^3+24 x^2+108 x+73$
• $y^2=7 x^6+64 x^5+111 x^4+21 x^3+92 x^2+62 x+62$
• $y^2=35 x^6+44 x^5+53 x^4+31 x^3+67 x^2+102 x+79$
• $y^2=x^6+25 x^5+92 x^4+43 x^3+79 x^2+10 x+82$
• $y^2=51 x^6+75 x^5+101 x^4+101 x^3+72 x^2+57 x+80$
• $y^2=18 x^6+43 x^5+41 x^4+101 x^3+13 x^2+69 x+61$
• $y^2=63 x^6+91 x^5+5 x^4+59 x^3+81 x^2+86 x+35$
• $y^2=58 x^6+82 x^5+36 x^4+89 x^3+29 x^2+29 x+40$
• $y^2=2 x^6+34 x^5+98 x^4+19 x^3+34 x^2+76 x+21$
• $y^2=39 x^6+83 x^5+7 x^4+93 x^3+34 x^2+13 x+42$
• $y^2=23 x^6+78 x^5+18 x^4+35 x^3+36 x^2+50 x+2$
• $y^2=23 x^6+104 x^5+65 x^4+65 x^3+31 x^2+34 x+16$
• $y^2=24 x^6+13 x^5+47 x^4+14 x^3+27 x^2+83 x+47$
• $y^2=5 x^6+57 x^5+103 x^4+35 x^3+69 x^2+68 x+39$
• $y^2=102 x^6+11 x^5+48 x^4+58 x^3+17 x^2+23$
• $y^2=74 x^6+94 x^5+30 x^4+47 x^3+7 x^2+32 x+83$
• $y^2=3 x^6+13 x^5+49 x^4+62 x^3+3 x^2+95 x+20$
• $y^2=65 x^6+94 x^5+107 x^4+104 x^3+42 x^2+50 x+11$
• $y^2=47 x^6+49 x^5+16 x^4+89 x^3+80 x^2+46 x+32$
• $y^2=66 x^6+28 x^5+73 x^4+52 x^3+90 x^2+11 x+43$
• $y^2=87 x^6+109 x^5+75 x^4+57 x^3+99 x^2+108 x+42$
• $y^2=63 x^5+3 x^4+7 x^3+42 x^2+49 x+15$
• $y^2=40 x^6+27 x^5+38 x^4+28 x^3+91 x^2+35 x+18$
• $y^2=20 x^6+80 x^5+41 x^4+75 x^3+111 x^2+89 x+28$
• $y^2=107 x^6+67 x^5+40 x^4+102 x^3+102 x^2+106 x+74$
• $y^2=27 x^6+33 x^5+85 x^4+78 x^3+55 x^2+27 x+73$
• $y^2=48 x^6+20 x^5+34 x^4+108 x^3+7 x^2+85 x+60$
• $y^2=107 x^6+85 x^5+51 x^4+62 x^3+50 x^2+29 x+51$
• $y^2=84 x^6+28 x^5+18 x^4+53 x^3+4 x^2+68 x+19$
• $y^2=59 x^6+60 x^5+83 x^4+2 x^3+52 x^2+5 x+6$
• $y^2=7 x^6+34 x^5+75 x^4+14 x^3+28 x^2+53 x+71$
• $y^2=50 x^6+65 x^5+65 x^4+12 x^3+64 x^2+30 x+58$
• $y^2=54 x^6+104 x^5+72 x^4+34 x^3+86 x^2+13 x+73$
• $y^2=69 x^6+112 x^5+106 x^4+98 x^3+16 x^2+2 x+30$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{113}$

The endomorphism algebra of this simple isogeny class is 4.0.2296445184.1.

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.u_ie	$2$	(not in LMFDB)