# Properties

 Label 2.13.ai_bh Base Field $\F_{13}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

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## Invariants

 Base field: $\F_{13}$ Dimension: $2$ L-polynomial: $( 1 - 7 x + 13 x^{2} )( 1 - x + 13 x^{2} )$ Frobenius angles: $\pm0.0772104791556$, $\pm0.455715642762$ 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 91 28665 4758208 802190025 137411044771 23309890007040 3938324793124267 665412853037812425 112454588112991214272 19005059294898370518825

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 6 172 2166 28084 370086 4829254 62763630 815726116 10604422158 137859184732

## Decomposition and endomorphism algebra

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