# Properties

 Label 2.13.aj_bo 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: $( 1 - 7 x + 13 x^{2} )( 1 - 2 x + 13 x^{2} )$ Frobenius angles: $\pm0.0772104791556$, $\pm0.410543812489$ Angle rank: $1$ (numerical) Jacobians: 4

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 4 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 84 28224 4826304 806190336 137254909764 23293210300416 3938419609170564 665462657105777664 112455406959154934976 19004944204233732354624

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 169 2198 28225 369665 4825798 62765141 815787169 10604499374 137858349889

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ah $\times$ 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:
Endomorphism algebra over $\overline{\F}_{13}$
 The base change of $A$ to $\F_{13^{6}}$ is 1.4826809.atm 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
All geometric endomorphisms are defined over $\F_{13^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{13^{2}}$  The base change of $A$ to $\F_{13^{2}}$ is 1.169.ax $\times$ 1.169.w. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{13^{3}}$  The base change of $A$ to $\F_{13^{3}}$ is 1.2197.acs $\times$ 1.2197.cs. 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.af_m $2$ 2.169.ab_agm 2.13.f_m $2$ 2.169.ab_agm 2.13.am_cj $3$ (not in LMFDB) 2.13.ad_q $3$ (not in LMFDB) 2.13.a_ax $3$ (not in LMFDB) 2.13.a_b $3$ (not in LMFDB) 2.13.a_w $3$ (not in LMFDB) 2.13.d_q $3$ (not in LMFDB) 2.13.j_bo $3$ (not in LMFDB) 2.13.m_cj $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.13.af_m $2$ 2.169.ab_agm 2.13.f_m $2$ 2.169.ab_agm 2.13.am_cj $3$ (not in LMFDB) 2.13.ad_q $3$ (not in LMFDB) 2.13.a_ax $3$ (not in LMFDB) 2.13.a_b $3$ (not in LMFDB) 2.13.a_w $3$ (not in LMFDB) 2.13.d_q $3$ (not in LMFDB) 2.13.j_bo $3$ (not in LMFDB) 2.13.m_cj $3$ (not in LMFDB) 2.13.ao_cx $6$ (not in LMFDB) 2.13.ak_bz $6$ (not in LMFDB) 2.13.ah_bk $6$ (not in LMFDB) 2.13.ae_be $6$ (not in LMFDB) 2.13.ac_aj $6$ (not in LMFDB) 2.13.a_w $6$ (not in LMFDB) 2.13.c_aj $6$ (not in LMFDB) 2.13.e_be $6$ (not in LMFDB) 2.13.h_bk $6$ (not in LMFDB) 2.13.k_bz $6$ (not in LMFDB) 2.13.o_cx $6$ (not in LMFDB) 2.13.a_aw $12$ (not in LMFDB) 2.13.a_ab $12$ (not in LMFDB) 2.13.a_x $12$ (not in LMFDB)