# Properties

 Label 2.49.av_hr 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: $1 - 21 x + 199 x^{2} - 1029 x^{3} + 2401 x^{4}$ Frobenius angles: $\pm0.0816995674884$, $\pm0.321155419757$ Angle rank: $2$ (numerical) Number field: 4.0.2427237.1 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1551 5662701 13863422727 33235938086373 79786477354344816 191577959731011790629 459985992090562697140623 1104427934737568361596936997 2651731058111204212095865353903 6366805834224365854473506703641856

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 29 2359 117839 5765323 282454754 13841050831 678222270083 33232938408019 1628413728252977 79792267216439614

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.2427237.1.
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_hr $2$ (not in LMFDB)