# Properties

 Label 2.139.abu_bfb Base Field $\F_{139}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{139}$ Dimension: $2$ L-polynomial: $( 1 - 23 x + 139 x^{2} )^{2}$ Frobenius angles: $\pm0.0707251543800$, $\pm0.0707251543800$ 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:

• $y^2=46x^6+32x^5+11x^4+65x^3+11x^2+32x+46$
• $y^2=x^6+78x^3+6$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 13689 363703041 7198725105936 139335482071482489 2692431221219247245049 52020850796219489993093376 1005095208705385769683771727889 19419444608054130384362092232563689 375203088588841496470422251715249251856 7249298872223650072661018639268040192724001

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 94 18820 2680468 373252324 51888440314 7212546884086 1002544366470286 139353667518166084 19370159749745440972 2692452204322731760900

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{139}$
 The isogeny class factors as 1.139.ax 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
All geometric endomorphisms are defined over $\F_{139}$.

## 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 2.139.a_ajr $2$ (not in LMFDB) 2.139.bu_bfb $2$ (not in LMFDB) 2.139.aq_en $3$ (not in LMFDB) 2.139.ah_adm $3$ (not in LMFDB) 2.139.o_mp $3$ (not in LMFDB) 2.139.x_pa $3$ (not in LMFDB) 2.139.bg_uo $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.139.a_ajr $2$ (not in LMFDB) 2.139.bu_bfb $2$ (not in LMFDB) 2.139.aq_en $3$ (not in LMFDB) 2.139.ah_adm $3$ (not in LMFDB) 2.139.o_mp $3$ (not in LMFDB) 2.139.x_pa $3$ (not in LMFDB) 2.139.bg_uo $3$ (not in LMFDB) 2.139.a_jr $4$ (not in LMFDB) 2.139.abn_yw $6$ (not in LMFDB) 2.139.abg_uo $6$ (not in LMFDB) 2.139.abe_qx $6$ (not in LMFDB) 2.139.ax_pa $6$ (not in LMFDB) 2.139.ao_mp $6$ (not in LMFDB) 2.139.aj_gk $6$ (not in LMFDB) 2.139.a_w $6$ (not in LMFDB) 2.139.a_iv $6$ (not in LMFDB) 2.139.h_adm $6$ (not in LMFDB) 2.139.j_gk $6$ (not in LMFDB) 2.139.q_en $6$ (not in LMFDB) 2.139.be_qx $6$ (not in LMFDB) 2.139.bn_yw $6$ (not in LMFDB) 2.139.a_aiv $12$ (not in LMFDB) 2.139.a_aw $12$ (not in LMFDB)