# Properties

 Label 2.13.ah_bg 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: $( 1 - 6 x + 13 x^{2} )( 1 - x + 13 x^{2} )$ Frobenius angles: $\pm0.187167041811$, $\pm0.455715642762$ Angle rank: $2$ (numerical) Jacobians: 6

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 6 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 104 31200 4954976 814320000 138011646344 23331990988800 3938757157931336 665396405968320000 112452136652458719584 19004887894305550380000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 7 185 2254 28513 371707 4833830 62770519 815705953 10604190982 137857941425

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ag $\times$ 1.13.ab 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.af_u $2$ 2.169.p_dk 2.13.f_u $2$ 2.169.p_dk 2.13.h_bg $2$ 2.169.p_dk
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.13.af_u $2$ 2.169.p_dk 2.13.f_u $2$ 2.169.p_dk 2.13.h_bg $2$ 2.169.p_dk 2.13.af_be $4$ (not in LMFDB) 2.13.ad_w $4$ (not in LMFDB) 2.13.d_w $4$ (not in LMFDB) 2.13.f_be $4$ (not in LMFDB)