# Properties

 Label 2.13.ah_bd 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: $1 - 7 x + 29 x^{2} - 91 x^{3} + 169 x^{4}$ Frobenius angles: $\pm0.138271059594$, $\pm0.479742145051$ Angle rank: $2$ (numerical) Number field: 4.0.72557.1 Galois group: $D_{4}$ Jacobians: 9

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 9 curves, and hence is principally polarizable:

• $y^2=2x^6+6x^5+2x^4+9x^3+7x+12$
• $y^2=9x^6+8x^5+10x^3+x^2+7x+2$
• $y^2=8x^6+6x^5+3x^4+11x^3+8x^2+6x+5$
• $y^2=8x^6+9x^5+6x^4+8x^3+10x^2+3x+1$
• $y^2=2x^6+2x^5+9x^3+11$
• $y^2=5x^5+12x^4+10x^2+12x+6$
• $y^2=8x^6+3x^5+5x^4+7x^3+8x^2+9x+7$
• $y^2=11x^6+2x^5+9x^4+4x^3+7x^2+7x+1$
• $y^2=12x^5+12x^3+12x^2+1$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 101 29997 4815377 808029189 137973330176 23336034433173 3938794045074017 665419950171347877 112455794370804724637 19005082979348421660672

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 7 179 2191 28291 371602 4834667 62771107 815734819 10604535907 137859356534

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The endomorphism algebra of this simple isogeny class is 4.0.72557.1.
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_bd $2$ 2.169.j_adr