Abelian variety isogeny class 2.197.acd_bsg over $\F_{197}$

• Label: 2.197.acd_bsg
• Base field: $\F_{197}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{197}$
Dimension:	$2$
L-polynomial:	$( 1 - 28 x + 197 x^{2} )( 1 - 27 x + 197 x^{2} )$
	$1 - 55 x + 1150 x^{2} - 10835 x^{3} + 38809 x^{4}$
Frobenius angles:	$\pm0.0226978709999$, $\pm0.0882242067266$
Angle rank:	$2$ (numerical)
Jacobians:	$0$

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$29070$	$1478209500$	$58381961470560$	$2268288826398000000$	$88036034924507367601350$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$143$	$38085$	$7636244$	$1506029393$	$296708059483$	$58451715804330$	$11514990365947519$	$2268453123228328513$	$446885265417814756148$	$88036397287440687536925$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{197}$.

Endomorphism algebra over $\F_{197}$

The isogeny class factors as 1.197.abc $\times$ 1.197.abb and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.197.abc : \(\Q(\sqrt{-1}) \).
• 1.197.abb : \(\Q(\sqrt{-59}) \).

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
2.197.ab_any	$2$	(not in LMFDB)
2.197.b_any	$2$	(not in LMFDB)
2.197.cd_bsg	$2$	(not in LMFDB)