# Properties

 Label 2.49.au_hk 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: $1 - 20 x + 192 x^{2} - 980 x^{3} + 2401 x^{4}$ Frobenius angles: $\pm0.151227437545$, $\pm0.318680514787$ Angle rank: $2$ (numerical) Number field: 4.0.3283200.1 Galois group: $D_{4}$ Jacobians: 16

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 16 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1594 5728836 13909691914 33259903165200 79796362518183754 191581217848125747396 459986798595930939118714 1104428036413689150823219200 2651731021732203448287443867194 6366805796707310078942043343032516

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 30 2386 118230 5769478 282489750 13841286226 678223459230 33232941467518 1628413705912830 79792266746255506

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.3283200.1.
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.u_hk $2$ (not in LMFDB)