# Properties

 Label 2.49.ax_it 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: $1 - 23 x + 227 x^{2} - 1127 x^{3} + 2401 x^{4}$ Frobenius angles: $\pm0.100880320937$, $\pm0.256439151204$ Angle rank: $2$ (numerical) Number field: 4.0.328029.1 Galois group: $D_{4}$ Jacobians: 4

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 4 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1479 5589141 13854847527 33252264620181 79798399975066224 191581894749761159229 459986317049419530286359 1104427610979578986417072389 2651730897254674257518743217727 6366805815027454591519116312274176

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 27 2327 117765 5768155 282496962 13841335127 678222749217 33232928665939 1628413629471855 79792266975853502

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.328029.1.
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.x_it $2$ (not in LMFDB)