# Properties

 Label 2.13.ai_bm 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: $( 1 - 6 x + 13 x^{2} )( 1 - 2 x + 13 x^{2} )$ Frobenius angles: $\pm0.187167041811$, $\pm0.410543812489$ Angle rank: $2$ (numerical) Jacobians: 12

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 12 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 96 30720 5025888 818380800 137854828896 23315295467520 3938851984386912 665446208805273600 112452955480771954272 19004772804678878361600

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 6 182 2286 28654 371286 4830374 62772030 815767006 10604268198 137857106582

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ag $\times$ 1.13.ac 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_o $2$ 2.169.m_eo 2.13.e_o $2$ 2.169.m_eo 2.13.i_bm $2$ 2.169.m_eo 2.13.al_ce $3$ (not in LMFDB) 2.13.b_aq $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.13.ae_o $2$ 2.169.m_eo 2.13.e_o $2$ 2.169.m_eo 2.13.i_bm $2$ 2.169.m_eo 2.13.al_ce $3$ (not in LMFDB) 2.13.b_aq $3$ (not in LMFDB) 2.13.ag_bi $4$ (not in LMFDB) 2.13.ac_s $4$ (not in LMFDB) 2.13.c_s $4$ (not in LMFDB) 2.13.g_bi $4$ (not in LMFDB) 2.13.an_cq $6$ (not in LMFDB) 2.13.ab_aq $6$ (not in LMFDB) 2.13.ab_ae $6$ (not in LMFDB) 2.13.b_ae $6$ (not in LMFDB) 2.13.l_ce $6$ (not in LMFDB) 2.13.n_cq $6$ (not in LMFDB) 2.13.al_cc $12$ (not in LMFDB) 2.13.aj_bu $12$ (not in LMFDB) 2.13.ad_ac $12$ (not in LMFDB) 2.13.ab_g $12$ (not in LMFDB) 2.13.b_g $12$ (not in LMFDB) 2.13.d_ac $12$ (not in LMFDB) 2.13.j_bu $12$ (not in LMFDB) 2.13.l_cc $12$ (not in LMFDB)