# Properties

 Label 2.13.ak_bv 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: $( 1 - 7 x + 13 x^{2} )( 1 - 3 x + 13 x^{2} )$ Frobenius angles: $\pm0.0772104791556$, $\pm0.363422825076$ Angle rank: $2$ (numerical) Jacobians: 2

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 2 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 77 27489 4868864 811722681 137321294957 23277766127616 3937639431868373 665478087042808809 112458102440893104896 19004999110047710270049

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 164 2218 28420 369844 4822598 62752708 815806084 10604753554 137858748164

## Decomposition and endomorphism algebra

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