# Properties

 Label 2.49.aw_if 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: $1 - 22 x + 213 x^{2} - 1078 x^{3} + 2401 x^{4}$ Frobenius angles: $\pm0.0895606637112$, $\pm0.290867746445$ Angle rank: $2$ (numerical) Number field: 4.0.1069632.1 Galois group: $D_{4}$ Jacobians: 8

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 8 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1515 5628225 13861940940 33245120238825 79792074188073075 191579344836288051600 459985860344438202800595 1104427717001232941391796425 2651731001171207920385266455180 6366805846821203728669412351030625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 28 2344 117826 5766916 282474568 13841150902 678222075832 33232931856196 1628413693286434 79792267374310024

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.1069632.1.
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.w_if $2$ (not in LMFDB)