# Properties

 Label 2.13.ai_bq 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: $( 1 - 4 x + 13 x^{2} )^{2}$ Frobenius angles: $\pm0.312832958189$, $\pm0.312832958189$ Angle rank: $1$ (numerical) Jacobians: 5

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 5 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 100 32400 5244100 829440000 137678102500 23258821107600 3935731667868100 665417390653440000 112459061652687456100 19005152104108790010000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 6 190 2382 29038 370806 4818670 62722302 815731678 10604844006 137859857950

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$
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.a_k $2$ 2.169.u_qw 2.13.i_bq $2$ 2.169.u_qw 2.13.e_d $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.13.a_k $2$ 2.169.u_qw 2.13.i_bq $2$ 2.169.u_qw 2.13.e_d $3$ (not in LMFDB) 2.13.am_ck $4$ (not in LMFDB) 2.13.ak_by $4$ (not in LMFDB) 2.13.ac_c $4$ (not in LMFDB) 2.13.a_ak $4$ (not in LMFDB) 2.13.c_c $4$ (not in LMFDB) 2.13.k_by $4$ (not in LMFDB) 2.13.m_ck $4$ (not in LMFDB) 2.13.ae_d $6$ (not in LMFDB) 2.13.a_ay $8$ (not in LMFDB) 2.13.a_y $8$ (not in LMFDB) 2.13.ag_x $12$ (not in LMFDB) 2.13.g_x $12$ (not in LMFDB)