Abelian variety isogeny class 2.109.abi_tj over $\F_{109}$

• Label: 2.109.abi_tj
• Base field: $\F_{109}$
• 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:	$\F_{109}$
Dimension:	$2$
L-polynomial:	$( 1 - 19 x + 109 x^{2} )( 1 - 15 x + 109 x^{2} )$
	$1 - 34 x + 503 x^{2} - 3706 x^{3} + 11881 x^{4}$
Frobenius angles:	$\pm0.136131296974$, $\pm0.244888641568$
Angle rank:	$2$ (numerical)
Jacobians:	$42$
Isomorphism classes:	72

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8645$	$139400625$	$1678246519040$	$19929441718265625$	$236741326284779040725$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$76$	$11732$	$1295914$	$141185188$	$15386561836$	$1677102532742$	$182803921971964$	$19925626408252228$	$2171893279319883346$	$236736367467278869652$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 42 curves (of which all are hyperelliptic):

• $y^2=28 x^6+34 x^5+91 x^4+67 x^3+51 x^2+61 x+3$
• $y^2=73 x^6+58 x^5+32 x^4+51 x^3+3 x^2+93 x+65$
• $y^2=29 x^6+7 x^5+63 x^4+72 x^3+103 x^2+8 x+101$
• $y^2=37 x^6+39 x^5+82 x^4+80 x^3+48 x^2+95 x+91$
• $y^2=40 x^6+97 x^5+62 x^4+8 x^3+92 x^2+76 x+107$
• $y^2=92 x^6+64 x^5+52 x^4+8 x^3+3 x^2+17 x+41$
• $y^2=95 x^6+76 x^5+54 x^4+86 x^3+65 x^2+53 x+53$
• $y^2=31 x^6+43 x^5+105 x^4+38 x^3+20 x^2+71 x+58$
• $y^2=63 x^6+45 x^5+13 x^4+92 x^3+90 x^2+80 x+4$
• $y^2=66 x^6+x^5+22 x^4+42 x^3+80 x^2+97 x+39$
• $y^2=6 x^6+51 x^5+84 x^4+101 x^3+89 x^2+37 x+24$
• $y^2=12 x^6+90 x^5+90 x^4+76 x^3+107 x^2+105 x+13$
• $y^2=55 x^6+60 x^5+24 x^4+64 x^3+56 x^2+66 x+70$
• $y^2=14 x^6+59 x^5+7 x^4+57 x^3+79 x^2+58 x+105$
• $y^2=74 x^6+19 x^5+38 x^4+108 x^3+12 x^2+65 x+73$
• $y^2=14 x^6+20 x^5+76 x^4+14 x^3+17 x^2+37 x+79$
• $y^2=57 x^6+8 x^5+108 x^4+81 x^3+63 x^2+60 x+16$
• $y^2=72 x^6+99 x^5+7 x^4+28 x^3+7 x^2+99 x+72$
• $y^2=103 x^6+46 x^5+39 x^4+30 x^3+107 x^2+94 x+40$
• $y^2=17 x^6+93 x^5+15 x^4+49 x^3+83 x^2+12 x+33$
• and

• $y^2=45 x^6+93 x^5+85 x^4+11 x^3+92 x^2+104 x+4$
• $y^2=28 x^6+15 x^5+103 x^4+67 x^3+17 x^2+75 x+3$
• $y^2=49 x^6+17 x^5+103 x^4+45 x^3+77 x^2+96 x+78$
• $y^2=90 x^6+71 x^5+x^4+102 x^3+72 x^2+77 x+76$
• $y^2=3 x^6+73 x^5+64 x^4+73 x^3+x^2+15 x+106$
• $y^2=5 x^6+87 x^5+63 x^4+89 x^3+48 x^2+41 x+39$
• $y^2=57 x^6+58 x^5+44 x^4+67 x^3+96 x^2+59 x+24$
• $y^2=55 x^6+59 x^5+35 x^4+45 x^3+26 x^2+8 x+19$
• $y^2=22 x^6+74 x^5+82 x^4+36 x^3+46 x^2+96 x+86$
• $y^2=4 x^6+15 x^5+36 x^4+59 x^3+78 x^2+65 x+88$
• $y^2=67 x^6+14 x^5+6 x^4+88 x^3+103 x^2+14 x+42$
• $y^2=86 x^6+2 x^5+102 x^4+26 x^3+43 x^2+51 x+107$
• $y^2=57 x^6+98 x^5+35 x^4+19 x^3+9 x^2+37 x+58$
• $y^2=17 x^6+81 x^5+6 x^4+71 x^3+95 x^2+72 x+53$
• $y^2=95 x^6+67 x^5+87 x^4+99 x^3+79 x^2+62 x+77$
• $y^2=105 x^6+77 x^5+22 x^4+5 x^3+69 x^2+9 x+48$
• $y^2=84 x^6+49 x^5+11 x^4+45 x^3+77 x^2+3 x+36$
• $y^2=65 x^6+75 x^5+70 x^4+23 x^3+86 x^2+22 x+50$
• $y^2=73 x^6+76 x^5+98 x^4+91 x^3+75 x^2+19 x+83$
• $y^2=52 x^6+54 x^5+8 x^4+7 x^3+98 x+51$
• $y^2=15 x^6+52 x^5+51 x^4+68 x^3+70 x^2+3 x+85$
• $y^2=59 x^6+82 x^5+88 x^4+100 x^3+97 x^2+29 x+18$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{109}$.

Endomorphism algebra over $\F_{109}$

The isogeny class factors as 1.109.at $\times$ 1.109.ap and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.109.at : \(\Q(\sqrt{-3}) \).
• 1.109.ap : \(\Q(\sqrt{-211}) \).

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
2.109.ae_acp	$2$	(not in LMFDB)
2.109.e_acp	$2$	(not in LMFDB)
2.109.bi_tj	$2$	(not in LMFDB)
2.109.an_hg	$3$	(not in LMFDB)
2.109.c_abl	$3$	(not in LMFDB)