# Properties

 Label 2.13.ai_bp 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: $( 1 - 5 x + 13 x^{2} )( 1 - 3 x + 13 x^{2} )$ Frobenius angles: $\pm0.256122854178$, $\pm0.363422825076$ Angle rank: $2$ (numerical) Jacobians: 4

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 4 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 99 31977 5189184 826829289 137766360699 23277766127616 3936722393153187 665415199774578825 112455591899559644736 19004941444519141523577

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 6 188 2358 28948 371046 4822598 62738094 815728996 10604516814 137858329868

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.af $\times$ 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.ac_l $2$ 2.169.s_nr 2.13.c_l $2$ 2.169.s_nr 2.13.i_bp $2$ 2.169.s_nr 2.13.af_bg $3$ (not in LMFDB) 2.13.e_f $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.13.ac_l $2$ 2.169.s_nr 2.13.c_l $2$ 2.169.s_nr 2.13.i_bp $2$ 2.169.s_nr 2.13.af_bg $3$ (not in LMFDB) 2.13.e_f $3$ (not in LMFDB) 2.13.ak_bv $6$ (not in LMFDB) 2.13.ae_f $6$ (not in LMFDB) 2.13.ab_u $6$ (not in LMFDB) 2.13.b_u $6$ (not in LMFDB) 2.13.f_bg $6$ (not in LMFDB) 2.13.k_bv $6$ (not in LMFDB)