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

• Label: 2.49.aw_ig
• 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:	$1 - 22 x + 214 x^{2} - 1078 x^{3} + 2401 x^{4}$
Frobenius angles:	$\pm0.105638774281$, $\pm0.284692968099$
Angle rank:	$2$ (numerical)
Number field:	4.0.62000.1
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	16

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1516$	$5633456$	$13869764236$	$33251361370880$	$79795367899657996$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$28$	$2346$	$117892$	$5767998$	$282486228$	$13841240106$	$678222551692$	$33232933343358$	$1628413693333228$	$79792267354598506$

Jacobians and polarizations

This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.62000.1.

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_ig	$2$	(not in LMFDB)