Abelian variety isogeny class 2.157.abu_bgk over $\F_{157}$

• Label: 2.157.abu_bgk
• Base field: $\F_{157}$
• 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:	$\F_{157}$
Dimension:	$2$
L-polynomial:	$( 1 - 24 x + 157 x^{2} )( 1 - 22 x + 157 x^{2} )$
	$1 - 46 x + 842 x^{2} - 7222 x^{3} + 24649 x^{4}$
Frobenius angles:	$\pm0.0929086555916$, $\pm0.158946998144$
Angle rank:	$2$ (numerical)
Jacobians:	$20$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$18224$	$597018240$	$14965221369392$	$369145834560921600$	$9099100440945786045104$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$112$	$24218$	$3867088$	$607574254$	$95389417552$	$14976080878826$	$2351243412634480$	$369145196213333086$	$57955795564572022576$	$9099059901172544402618$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

• $y^2=19 x^6+151 x^5+19 x^4+62 x^3+117 x^2+73 x+68$
• $y^2=125 x^6+67 x^5+38 x^4+12 x^3+38 x^2+67 x+125$
• $y^2=30 x^6+18 x^5+93 x^4+102 x^3+42 x^2+70 x+51$
• $y^2=104 x^6+54 x^5+32 x^4+37 x^3+32 x^2+54 x+104$
• $y^2=2 x^6+146 x^5+150 x^4+85 x^3+62 x^2+31 x+54$
• $y^2=133 x^6+47 x^5+142 x^4+19 x^3+23 x^2+17 x+24$
• $y^2=22 x^6+64 x^5+18 x^4+71 x^3+6 x^2+42 x+152$
• $y^2=149 x^6+21 x^5+153 x^4+17 x^3+153 x^2+21 x+149$
• $y^2=140 x^6+10 x^5+68 x^4+87 x^3+68 x^2+10 x+140$
• $y^2=72 x^6+155 x^5+13 x^4+97 x^3+13 x^2+155 x+72$
• $y^2=7 x^6+70 x^5+62 x^4+147 x^3+62 x^2+70 x+7$
• $y^2=6 x^6+97 x^5+40 x^4+54 x^3+40 x^2+97 x+6$
• $y^2=150 x^6+14 x^5+27 x^4+79 x^3+9 x^2+19 x+23$
• $y^2=142 x^6+91 x^5+156 x^4+30 x^3+100 x^2+28 x+63$
• $y^2=101 x^6+16 x^5+107 x^4+87 x^3+7 x^2+144 x+58$
• $y^2=149 x^6+56 x^5+85 x^4+92 x^3+85 x^2+56 x+149$
• $y^2=85 x^6+145 x^5+3 x^4+7 x^3+3 x^2+145 x+85$
• $y^2=96 x^6+136 x^5+121 x^4+83 x^3+75 x^2+128 x+135$
• $y^2=13 x^6+74 x^5+89 x^4+94 x^3+13 x^2+70 x+52$
• $y^2=125 x^6+147 x^5+131 x^4+110 x^3+131 x^2+147 x+125$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{157}$.

Endomorphism algebra over $\F_{157}$

The isogeny class factors as 1.157.ay $\times$ 1.157.aw and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.157.ay : \(\Q(\sqrt{-13}) \).
• 1.157.aw : \(\Q(\sqrt{-1}) \).

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
2.157.ac_aig	$2$	(not in LMFDB)
2.157.c_aig	$2$	(not in LMFDB)
2.157.bu_bgk	$2$	(not in LMFDB)