# Properties

 Label 2.13.aj_bp 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: $1 - 9 x + 41 x^{2} - 117 x^{3} + 169 x^{4}$ Frobenius angles: $\pm0.109149799241$, $\pm0.400911184348$ Angle rank: $2$ (numerical) Number field: 4.0.122157.1 Galois group: $D_{4}$ Jacobians: 2

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 2 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 85 28645 4886905 810796725 137471390800 23301803950765 3938925735789745 665496620891533125 112456859467929137365 19004959616264376217600

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 171 2225 28387 370250 4827579 62773205 815828803 10604636345 137858461686

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The endomorphism algebra of this simple isogeny class is 4.0.122157.1.
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.j_bp $2$ 2.169.b_adj