Abelian variety isogeny class 2.49.aw_if over $\F_{7^{2}}$

• Label: 2.49.aw_if
• Base field: $\F_{7^{2}}$
• 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:	$\F_{7^{2}}$
Dimension:	$2$
L-polynomial:	$1 - 22 x + 213 x^{2} - 1078 x^{3} + 2401 x^{4}$
Frobenius angles:	$\pm0.0895606637112$, $\pm0.290867746445$
Angle rank:	$2$ (numerical)
Number field:	4.0.1069632.1
Galois group:	$D_{4}$
Jacobians:	$8$
Isomorphism classes:	8

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1515$	$5628225$	$13861940940$	$33245120238825$	$79792074188073075$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$28$	$2344$	$117826$	$5766916$	$282474568$	$13841150902$	$678222075832$	$33232931856196$	$1628413693286434$	$79792267374310024$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=(5a+1)x^6+5ax^5+(6a+4)x^4+(6a+3)x^3+(3a+4)x^2+(4a+2)x+3a+3$
• $y^2=(3a+4)x^6+6x^5+(4a+6)x^4+(5a+1)x^3+(a+2)x^2+4x+5a+5$
• $y^2=(3a+4)x^6+3ax^5+2ax^4+(5a+6)x^3+(3a+6)x^2+(a+6)x+3a+5$
• $y^2=(5a+2)x^6+(6a+6)x^5+(6a+3)x^4+(3a+3)x^3+(a+3)x^2+(4a+3)x+4a+6$
• $y^2=6x^6+4x^5+(4a+4)x^4+(2a+6)x^3+(4a+2)x^2+(4a+6)x+4a+3$
• $y^2=(2a+1)x^6+ax^5+(6a+1)x^4+(a+3)x^3+(3a+3)x^2+(3a+3)x+3a+1$
• $y^2=(a+3)x^6+(3a+3)x^5+(4a+2)x^4+(4a+2)x^3+(3a+6)x+2$
• $y^2=(2a+1)x^6+6x^5+(5a+3)x^4+(2a+5)x^3+(4a+2)x^2+(a+4)x+5a$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7^{2}}$.

Endomorphism algebra over $\F_{7^{2}}$

The endomorphism algebra of this simple isogeny class is 4.0.1069632.1.

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
2.49.w_if	$2$	(not in LMFDB)