# Properties

 Label 2.49.aw_ig Base Field $\F_{7^{2}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ 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

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

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

## Point counts

This isogeny class contains the Jacobians of 12 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1516 5633456 13869764236 33251361370880 79795367899657996 191580579525239810096 459986183082862858304236 1104427766423952082185236480 2651731001247407224545011991916 6366805845248377024018375311092016

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

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.62000.1.
All geometric endomorphisms are defined over $\F_{7^{2}}$.

## 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)