# Properties

 Label 2.13.aj_bs 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: $( 1 - 6 x + 13 x^{2} )( 1 - 3 x + 13 x^{2} )$ Frobenius angles: $\pm0.187167041811$, $\pm0.363422825076$ Angle rank: $2$ (numerical) Jacobians: 3

This isogeny class is not 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 3 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 88 29920 5070208 823996800 137921504248 23299836651520 3938071721433784 665461638360921600 112455650903749808512 19004827709997678181600

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 177 2306 28849 371465 4827174 62759597 815785921 10604522378 137857504857

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ag $\times$ 1.13.ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{13}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.13.ad_i $2$ 2.169.h_gm 2.13.d_i $2$ 2.169.h_gm 2.13.j_bs $2$ 2.169.h_gm
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.13.ad_i $2$ 2.169.h_gm 2.13.d_i $2$ 2.169.h_gm 2.13.j_bs $2$ 2.169.h_gm 2.13.ah_bm $4$ (not in LMFDB) 2.13.ab_o $4$ (not in LMFDB) 2.13.b_o $4$ (not in LMFDB) 2.13.h_bm $4$ (not in LMFDB)