Invariants
Base field: | $\F_{139}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 23 x + 139 x^{2} )( 1 - 19 x + 139 x^{2} )$ |
$1 - 42 x + 715 x^{2} - 5838 x^{3} + 19321 x^{4}$ | |
Frobenius angles: | $\pm0.0707251543800$, $\pm0.201746658314$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $32$ |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $14157$ | $366906969$ | $7208491386096$ | $139356427372718985$ | $2692465343398114418277$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $98$ | $18988$ | $2684108$ | $373308436$ | $51889097918$ | $7212552387766$ | $1002544384691402$ | $139353667103273956$ | $19370159738716790612$ | $2692452204156214591228$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 32 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=26x^6+124x^5+108x^4+96x^3+108x^2+124x+26$
- $y^2=43x^6+18x^5+74x^4+20x^3+3x^2+24x+109$
- $y^2=103x^6+44x^5+129x^4+71x^3+46x^2+42x+126$
- $y^2=22x^6+118x^5+49x^4+60x^3+49x^2+118x+22$
- $y^2=71x^6+90x^5+120x^4+17x^3+58x^2+16x+61$
- $y^2=127x^6+108x^5+59x^4+29x^3+130x^2+79x+123$
- $y^2=99x^6+27x^5+9x^4+127x^3+9x^2+27x+99$
- $y^2=114x^6+114x^5+77x^4+33x^3+77x^2+114x+114$
- $y^2=50x^6+59x^5+120x^4+32x^3+25x^2+8x+8$
- $y^2=17x^6+106x^5+45x^4+91x^3+45x^2+106x+17$
- $y^2=7x^6+60x^5+27x^4+41x^3+27x^2+60x+7$
- $y^2=62x^6+103x^5+6x^4+5x^3+116x^2+115x+71$
- $y^2=123x^6+66x^4+112x^3+60x^2+78x+37$
- $y^2=76x^6+97x^5+3x^4+66x^3+21x^2+95x+40$
- $y^2=86x^6+65x^5+26x^4+79x^3+9x^2+94x+16$
- $y^2=134x^6+42x^5+90x^4+8x^3+90x^2+42x+134$
- $y^2=35x^6+98x^5+5x^4+115x^3+102x^2+28x+94$
- $y^2=122x^6+102x^5+52x^4+122x^3+52x^2+102x+122$
- $y^2=84x^6+54x^5+108x^4+3x^3+108x^2+54x+84$
- $y^2=93x^6+61x^5+80x^4+111x^3+80x^2+61x+93$
- and 12 more
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{139}$.
Endomorphism algebra over $\F_{139}$The isogeny class factors as 1.139.ax $\times$ 1.139.at and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.