Abelian variety isogeny class 2.17.aj_bt over $\F_{17}$

• Label: 2.17.aj_bt
• Base field: $\F_{17}$
• 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:	$\F_{17}$
Dimension:	$2$
L-polynomial:	$1 - 9 x + 45 x^{2} - 153 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.132563239740$, $\pm0.443398416497$
Angle rank:	$2$ (numerical)
Number field:	4.0.1003477.1
Galois group:	$D_{4}$
Jacobians:	$3$
Isomorphism classes:	3

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$173$	$85981$	$24276917$	$6943911541$	$2015023398608$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$299$	$4941$	$83139$	$1419174$	$24150179$	$410416533$	$6975896803$	$118587753945$	$2015994794654$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=6x^6+16x^5+7x^4+x^3+12x^2+9x+5$
• $y^2=11x^6+3x^5+x^4+8x^3+10x^2+7x+6$
• $y^2=3x^6+8x^5+13x^3+9x^2+11x+14$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{17}$.

Endomorphism algebra over $\F_{17}$

The endomorphism algebra of this simple isogeny class is 4.0.1003477.1.

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
2.17.j_bt	$2$	(not in LMFDB)