# Properties

 Label 2.13.ah_ba Base Field $\F_{13}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{13}$ Dimension: $2$ L-polynomial: $( 1 - 7 x + 13 x^{2} )( 1 + 13 x^{2} )$ Frobenius angles: $\pm0.0772104791556$, $\pm0.5$ Angle rank: $1$ (numerical) Jacobians: 2

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1]$

## Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 98 28812 4677344 800743104 137700691898 23316859190016 3937502572968986 665386850963116800 112456662227757292256 19005143383543652756652

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 7 173 2128 28033 370867 4830698 62750527 815694241 10604617744 137859794693

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ah $\times$ 1.13.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{13}$
 The base change of $A$ to $\F_{13^{2}}$ is 1.169.ax $\times$ 1.169.ba. The endomorphism algebra for each factor is: 1.169.ax : $$\Q(\sqrt{-3})$$. 1.169.ba : the quaternion algebra over $$\Q$$ ramified at $13$ and $\infty$.
All geometric endomorphisms are defined over $\F_{13^{2}}$.

## 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.h_ba $2$ 2.169.d_aka 2.13.c_ba $3$ (not in LMFDB) 2.13.f_ba $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.13.h_ba $2$ 2.169.d_aka 2.13.c_ba $3$ (not in LMFDB) 2.13.f_ba $3$ (not in LMFDB) 2.13.af_ba $6$ (not in LMFDB) 2.13.ac_ba $6$ (not in LMFDB)