# Properties

 Label 2.13.ak_bz 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: $( 1 - 5 x + 13 x^{2} )^{2}$ Frobenius angles: $\pm0.256122854178$, $\pm0.256122854178$ Angle rank: $1$ (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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 81 29241 5143824 835152201 138435340761 23293210300416 3935794779943569 665324522896775625 112452896477996048016 19005002916235392383721

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 172 2338 29236 372844 4825798 62723308 815617828 10604262634 137858775772

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
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_b $2$ 2.169.c_nb 2.13.k_bz $2$ 2.169.c_nb 2.13.ah_bk $3$ (not in LMFDB) 2.13.ae_be $3$ (not in LMFDB) 2.13.c_aj $3$ (not in LMFDB) 2.13.f_m $3$ (not in LMFDB) 2.13.o_cx $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.13.a_b $2$ 2.169.c_nb 2.13.k_bz $2$ 2.169.c_nb 2.13.ah_bk $3$ (not in LMFDB) 2.13.ae_be $3$ (not in LMFDB) 2.13.c_aj $3$ (not in LMFDB) 2.13.f_m $3$ (not in LMFDB) 2.13.o_cx $3$ (not in LMFDB) 2.13.a_ab $4$ (not in LMFDB) 2.13.ao_cx $6$ (not in LMFDB) 2.13.am_cj $6$ (not in LMFDB) 2.13.aj_bo $6$ (not in LMFDB) 2.13.af_m $6$ (not in LMFDB) 2.13.ad_q $6$ (not in LMFDB) 2.13.ac_aj $6$ (not in LMFDB) 2.13.a_ax $6$ (not in LMFDB) 2.13.a_w $6$ (not in LMFDB) 2.13.d_q $6$ (not in LMFDB) 2.13.e_be $6$ (not in LMFDB) 2.13.h_bk $6$ (not in LMFDB) 2.13.j_bo $6$ (not in LMFDB) 2.13.m_cj $6$ (not in LMFDB) 2.13.a_aw $12$ (not in LMFDB) 2.13.a_x $12$ (not in LMFDB)