# Properties

 Label 2.13.ah_be Base Field $\F_{13}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{13}$ Dimension: $2$ L-polynomial: $1 - 7 x + 30 x^{2} - 91 x^{3} + 169 x^{4}$ Frobenius angles: $\pm0.155060670890$, $\pm0.472256355230$ Angle rank: $2$ (numerical) Number field: 4.0.640332.1 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:

• $y^2=6x^6+3x^5+10x^4+4x^3+3x^2+x+10$
• $y^2=6x^6+4x^5+6x^4+12x$
• $y^2=5x^6+7x^5+3x^4+x^3+8x^2+10x$
• $y^2=8x^6+9x^5+11x^4+8x^3+11x^2+10x+6$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 102 30396 4861728 810235776 138012101742 23337188957952 3938904764517342 665414807044689408 112454567999145347616 19005013455302895068316

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 7 181 2212 28369 371707 4834906 62772871 815728513 10604420260 137858852221

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The endomorphism algebra of this simple isogeny class is 4.0.640332.1.
All geometric endomorphisms are defined over $\F_{13}$.

## 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 2.13.h_be $2$ 2.169.l_abk