# Properties

 Label 2.49.av_hq 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: $1 - 21 x + 198 x^{2} - 1029 x^{3} + 2401 x^{4}$ Frobenius angles: $\pm0.0658419374597$, $\pm0.325440921041$ Angle rank: $2$ (numerical) Number field: 4.0.1983580.1 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1550 5657500 13855964600 33230344600000 79783719103742750 191576956129185280000 459985682426761289771150 1104427828823344830741600000 2651731011712694383487262270200 6366805813860283284355398718937500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 29 2357 117776 5764353 282444989 13840978322 678221813501 33232935220993 1628413699759904 79792266961225877

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.1983580.1.
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.v_hq $2$ (not in LMFDB)