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

• Label: 2.157.abu_bgh
• 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:	$( 1 - 25 x + 157 x^{2} )( 1 - 21 x + 157 x^{2} )$
	$1 - 46 x + 839 x^{2} - 7222 x^{3} + 24649 x^{4}$
Frobenius angles:	$\pm0.0220179720414$, $\pm0.183729987215$
Angle rank:	$2$ (numerical)
Jacobians:	$4$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$18221$	$596865297$	$14963616087056$	$369136533867138009$	$9099061476136699522301$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$112$	$24212$	$3866674$	$607558948$	$95389009072$	$14976072091622$	$2351243252077552$	$369145193654555716$	$57955795528588253338$	$9099059900726078728532$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

• $y^2=119 x^6+21 x^5+102 x^4+21 x^3+72 x^2+80 x+6$
• $y^2=150 x^6+137 x^5+125 x^4+113 x^3+9 x^2+8 x+115$
• $y^2=69 x^6+25 x^5+53 x^4+47 x^3+126 x^2+35 x+13$
• $y^2=96 x^6+16 x^5+76 x^4+84 x^3+76 x^2+16 x+96$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{157}$

The isogeny class factors as 1.157.az $\times$ 1.157.av 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.az : \(\Q(\sqrt{-3}) \).
• 1.157.av : \(\Q(\sqrt{-187}) \).

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.ae_aid	$2$	(not in LMFDB)
2.157.e_aid	$2$	(not in LMFDB)
2.157.bu_bgh	$2$	(not in LMFDB)
2.157.ak_df	$3$	(not in LMFDB)
2.157.ah_u	$3$	(not in LMFDB)