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

• Label: 2.49.aw_ij
• Base field: $\F_{7^{2}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• 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:	$1 - 22 x + 217 x^{2} - 1078 x^{3} + 2401 x^{4}$
Frobenius angles:	$\pm0.152971575916$, $\pm0.259933654116$
Angle rank:	$2$ (numerical)
Number field:	4.0.5696.1
Galois group:	$D_{4}$
Jacobians:	$2$

This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1519$	$5649161$	$13893241852$	$33269948476121$	$79804876484494399$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$28$	$2352$	$118090$	$5771220$	$282519888$	$13841470902$	$678223289968$	$33232928193444$	$1628413589373130$	$79792266418406752$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.5696.1.

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_ij	$2$	(not in LMFDB)