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

• Label: 2.49.aw_ik
• Base field: $\F_{7^{2}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{7^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 12 x + 49 x^{2} )( 1 - 10 x + 49 x^{2} )$
	$1 - 22 x + 218 x^{2} - 1078 x^{3} + 2401 x^{4}$
Frobenius angles:	$\pm0.172237328522$, $\pm0.246751714429$
Angle rank:	$2$ (numerical)
Jacobians:	$16$
Isomorphism classes:	48

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1520$	$5654400$	$13901070320$	$33276098764800$	$79807921836143600$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$28$	$2354$	$118156$	$5772286$	$282530668$	$13841535602$	$678223309372$	$33232923393406$	$1628413523188924$	$79792265876711474$

Jacobians and polarizations

This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

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

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

The isogeny class factors as 1.49.am $\times$ 1.49.ak and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.49.am : \(\Q(\sqrt{-13}) \).
• 1.49.ak : \(\Q(\sqrt{-6}) \).

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.ac_aw	$2$	(not in LMFDB)
2.49.c_aw	$2$	(not in LMFDB)
2.49.w_ik	$2$	(not in LMFDB)