# Properties

 Label 2.13.ak_by 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: $( 1 - 6 x + 13 x^{2} )( 1 - 4 x + 13 x^{2} )$ Frobenius angles: $\pm0.187167041811$, $\pm0.312832958189$ 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 80 28800 5074640 829440000 138211672400 23298078211200 3937112223300560 665417390653440000 112456038027485859920 19004963774689959120000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 170 2308 29038 372244 4826810 62744308 815731678 10604558884 137858491850

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ag $\times$ 1.13.ae and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{13}$
 The base change of $A$ to $\F_{13^{4}}$ is 1.28561.je 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$
All geometric endomorphisms are defined over $\F_{13^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{13^{2}}$  The base change of $A$ to $\F_{13^{2}}$ is 1.169.ak $\times$ 1.169.k. The endomorphism algebra for each factor is:

## 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_c $2$ 2.169.a_je 2.13.c_c $2$ 2.169.a_je 2.13.k_by $2$ 2.169.a_je
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.13.ac_c $2$ 2.169.a_je 2.13.c_c $2$ 2.169.a_je 2.13.k_by $2$ 2.169.a_je 2.13.am_ck $4$ (not in LMFDB) 2.13.ai_bq $4$ (not in LMFDB) 2.13.a_ak $4$ (not in LMFDB) 2.13.a_k $4$ (not in LMFDB) 2.13.i_bq $4$ (not in LMFDB) 2.13.m_ck $4$ (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.ae_d $12$ (not in LMFDB) 2.13.e_d $12$ (not in LMFDB) 2.13.g_x $12$ (not in LMFDB)