# Properties

 Label 2.49.au_he 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: $1 - 20 x + 186 x^{2} - 980 x^{3} + 2401 x^{4}$ Frobenius angles: $\pm0.0883562958487$, $\pm0.345388837514$ Angle rank: $2$ (numerical) Number field: 4.0.81216.1 Galois group: $D_{4}$ Jacobians: 28

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 28 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1588 5697744 13867090900 33230701630464 79784329442455828 191578366201333410000 459986657310244533256948 1104428187943134012864282624 2651731074872682735296972058100 6366805809512704005975291774565584

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 30 2374 117870 5764414 282447150 13841080198 678223250910 33232946027134 1628413738546110 79792266906739654

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.81216.1.
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_he $2$ (not in LMFDB)