# Properties

 Label 2.139.abn_yw 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: $( 1 - 23 x + 139 x^{2} )( 1 - 16 x + 139 x^{2} )$ Frobenius angles: $\pm0.0707251543800$, $\pm0.262608178953$ Angle rank: $1$ (numerical) Jacobians: 38

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 38 curves, and hence is principally polarizable:

• $y^2=43x^6+10x^5+82x^4+41x^3+55x^2+138x+101$
• $y^2=4x^6+20x^5+51x^4+21x^3+79x^2+93x+18$
• $y^2=82x^6+104x^5+46x^4+75x^3+112x^2+25x+32$
• $y^2=40x^6+4x^5+126x^4+107x^3+120x^2+126x+104$
• $y^2=52x^6+128x^5+92x^4+13x^3+63x^2+17x+109$
• $y^2=103x^6+48x^5+123x^4+61x^3+47x^2+27x+135$
• $y^2=62x^6+80x^5+134x^4+115x^3+83x^2+99x+75$
• $y^2=37x^6+53x^5+35x^4+120x^3+17x^2+26x+85$
• $y^2=114x^6+37x^5+12x^4+44x^3+138x^2+12x+53$
• $y^2=120x^6+74x^5+68x^4+91x^3+26x^2+3x+49$
• $y^2=112x^6+51x^5+95x^4+44x^3+29x^2+107x+137$
• $y^2=73x^6+6x^5+26x^4+82x^3+87x^2+10x+69$
• $y^2=17x^6+10x^5+79x^4+38x^3+121x^2+124x+41$
• $y^2=18x^6+107x^5+85x^4+103x^3+75x^2+58x+133$
• $y^2=108x^6+30x^5+85x^4+90x^3+40x^2+97x+24$
• $y^2=98x^6+135x^5+103x^4+97x^3+125x^2+29x+69$
• $y^2=32x^6+5x^5+48x^4+85x^3+30x^2+29x+6$
• $y^2=106x^6+81x^5+6x^4+75x^3+56x^2+x+91$
• $y^2=27x^6+33x^5+48x^4+59x^3+86x^2+23x+12$
• $y^2=x^5+68x^4+89x^3+88x^2+136x+104$
• and 18 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 14508 368909424 7212548148624 139358818209873600 2692454812753783550148 52020850796219489993093376 1005095154221553703409927785908 19419444487837611884973222276422400 375203088447024628530509907710222277264 7249298872162111218021873908096346034366704

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 101 19093 2685620 373314841 51888894971 7212546884086 1002544312124729 139353666655494001 19370159742424031660 2692452204299875698853

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{139}$
 The isogeny class factors as 1.139.ax $\times$ 1.139.aq 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 $\overline{\F}_{139}$
 The base change of $A$ to $\F_{139^{6}}$ is 1.7212549413161.actyqc 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
All geometric endomorphisms are defined over $\F_{139^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{139^{2}}$  The base change of $A$ to $\F_{139^{2}}$ is 1.19321.ajr $\times$ 1.19321.w. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{139^{3}}$  The base change of $A$ to $\F_{139^{3}}$ is 1.2685619.advc $\times$ 1.2685619.dvc. 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 2.139.ah_adm $2$ (not in LMFDB) 2.139.h_adm $2$ (not in LMFDB) 2.139.bn_yw $2$ (not in LMFDB) 2.139.abe_qx $3$ (not in LMFDB) 2.139.aj_gk $3$ (not in LMFDB) 2.139.a_ajr $3$ (not in LMFDB) 2.139.a_w $3$ (not in LMFDB) 2.139.a_iv $3$ (not in LMFDB) 2.139.j_gk $3$ (not in LMFDB) 2.139.be_qx $3$ (not in LMFDB) 2.139.bn_yw $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.139.ah_adm $2$ (not in LMFDB) 2.139.h_adm $2$ (not in LMFDB) 2.139.bn_yw $2$ (not in LMFDB) 2.139.abe_qx $3$ (not in LMFDB) 2.139.aj_gk $3$ (not in LMFDB) 2.139.a_ajr $3$ (not in LMFDB) 2.139.a_w $3$ (not in LMFDB) 2.139.a_iv $3$ (not in LMFDB) 2.139.j_gk $3$ (not in LMFDB) 2.139.be_qx $3$ (not in LMFDB) 2.139.bn_yw $3$ (not in LMFDB) 2.139.abu_bfb $6$ (not in LMFDB) 2.139.abg_uo $6$ (not in LMFDB) 2.139.ax_pa $6$ (not in LMFDB) 2.139.aq_en $6$ (not in LMFDB) 2.139.ao_mp $6$ (not in LMFDB) 2.139.o_mp $6$ (not in LMFDB) 2.139.q_en $6$ (not in LMFDB) 2.139.x_pa $6$ (not in LMFDB) 2.139.bg_uo $6$ (not in LMFDB) 2.139.bu_bfb $6$ (not in LMFDB) 2.139.a_aiv $12$ (not in LMFDB) 2.139.a_aw $12$ (not in LMFDB) 2.139.a_jr $12$ (not in LMFDB)