# Properties

 Label 2.13.ai_bl 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: $1 - 8 x + 37 x^{2} - 104 x^{3} + 169 x^{4}$ Frobenius angles: $\pm0.167453355204$, $\pm0.421338968608$ Angle rank: $2$ (numerical) Number field: 4.0.16025.1 Galois group: $D_{4}$ Jacobians: 10

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 10 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 95 30305 4971920 815356025 137825337375 23321168625920 3939210486334295 665457517069810025 112453624060644805520 19004830197396067022625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 6 180 2262 28548 371206 4831590 62777742 815780868 10604331246 137857522900

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The endomorphism algebra of this simple isogeny class is 4.0.16025.1.
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.i_bl $2$ 2.169.k_br