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

• Label: 2.197.aca_bpb
• Base field: $\F_{197}$
• 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:	$\F_{197}$
Dimension:	$2$
L-polynomial:	$1 - 52 x + 1067 x^{2} - 10244 x^{3} + 38809 x^{4}$
Frobenius angles:	$\pm0.0495399438151$, $\pm0.167628951893$
Angle rank:	$2$ (numerical)
Number field:	4.0.542736.2
Galois group:	$D_{4}$
Jacobians:	$4$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$29581$	$1484167513$	$58414372764100$	$2268417823589269369$	$88036449421385648571421$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$146$	$38240$	$7640486$	$1506115044$	$296709456466$	$58451734502750$	$11514990561085738$	$2268453124427693764$	$446885265409643759822$	$88036397287015566495200$

Jacobians and polarizations

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

• $y^2=16 x^6+146 x^5+10 x^4+196 x^3+76 x^2+26 x+41$
• $y^2=91 x^6+94 x^5+183 x^4+9 x^3+35 x^2+111 x+46$
• $y^2=14 x^6+87 x^5+132 x^4+6 x^3+41 x^2+169 x+9$
• $y^2=114 x^6+88 x^5+144 x^4+104 x^3+131 x^2+64 x+188$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{197}$

The endomorphism algebra of this simple isogeny class is 4.0.542736.2.

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
2.197.ca_bpb	$2$	(not in LMFDB)