# Properties

 Label 2.13.ag_t 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: $( 1 - 7 x + 13 x^{2} )( 1 + x + 13 x^{2} )$ Frobenius angles: $\pm0.0772104791556$, $\pm0.544284357238$ Angle rank: $2$ (numerical) Jacobians: 8

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 8 curves, and hence is principally polarizable:

• $y^2=7x^6+3x^5+3x+7$
• $y^2=2x^6+11x^5+x^4+3x^3+6x^2+5x+2$
• $y^2=8x^6+2x^5+7x^4+9x^3+4x^2+12x+9$
• $y^2=8x^6+5x^5+10x^4+9x^3+4x^2+11$
• $y^2=2x^6+4x^5+8x^4+x^3+11x^2+5x+11$
• $y^2=5x^6+x^5+9x^4+9x^2+3x+1$
• $y^2=6x^6+3x^5+12x^4+11x^3+12x^2+6x+7$
• $y^2=2x^6+9x^5+6x^4+5x^3+8x^2+3x+11$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 105 28665 4596480 802190025 137990339025 23309890007040 3936680352813705 665412853037812425 112458736342523370240 19005059294898370518825

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 8 172 2090 28084 371648 4829254 62737424 815726116 10604813330 137859184732

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ah $\times$ 1.13.b 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.ai_bh $2$ 2.169.c_ajd 2.13.g_t $2$ 2.169.c_ajd 2.13.i_bh $2$ 2.169.c_ajd 2.13.d_bc $3$ (not in LMFDB) 2.13.g_bf $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.13.ai_bh $2$ 2.169.c_ajd 2.13.g_t $2$ 2.169.c_ajd 2.13.i_bh $2$ 2.169.c_ajd 2.13.d_bc $3$ (not in LMFDB) 2.13.g_bf $3$ (not in LMFDB) 2.13.ag_bf $6$ (not in LMFDB) 2.13.ae_v $6$ (not in LMFDB) 2.13.ad_bc $6$ (not in LMFDB) 2.13.ab_y $6$ (not in LMFDB) 2.13.b_y $6$ (not in LMFDB) 2.13.e_v $6$ (not in LMFDB)