# Properties

 Label 2.49.au_hf 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: $1 - 20 x + 187 x^{2} - 980 x^{3} + 2401 x^{4}$ Frobenius angles: $\pm0.0998650046166$, $\pm0.341585099009$ Angle rank: $2$ (numerical) Number field: 4.0.5469200.1 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1589 5702921 13874188244 33235625576825 79786476121393629 191579012411680107536 459986815688615935867469 1104428239264561746233558825 2651731098559946289403790934164 6366805817766065145782928634039641

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 30 2376 117930 5765268 282454750 13841126886 678223484430 33232947571428 1628413753092330 79792267010175256

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.5469200.1.
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_hf $2$ (not in LMFDB)