Invariants
| Base field: | $\F_{139}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 23 x + 139 x^{2} )( 1 - 18 x + 139 x^{2} )$ |
| $1 - 41 x + 692 x^{2} - 5699 x^{3} + 19321 x^{4}$ | |
| Frobenius angles: | $\pm0.0707251543800$, $\pm0.223543330897$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $8$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $14274$ | $367612596$ | $7210128042936$ | $139358209022326944$ | $2692463754100883071374$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $99$ | $19025$ | $2684718$ | $373313209$ | $51889067289$ | $7212550717586$ | $1002544354646451$ | $139353666777378289$ | $19370159737288567722$ | $2692452204183174022625$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=85x^6+123x^5+99x^4+110x^3+83x^2+58x+7$
- $y^2=3x^6+87x^5+101x^4+35x^3+75x^2+71x+108$
- $y^2=38x^6+75x^5+118x^4+80x^3+70x^2+4x+57$
- $y^2=132x^6+126x^5+122x^4+94x^3+120x^2+5x+13$
- $y^2=29x^6+122x^5+31x^4+132x^3+x^2+123x+136$
- $y^2=105x^6+19x^5+51x^4+8x^3+118x^2+15x+29$
- $y^2=99x^6+8x^5+11x^4+6x^3+14x^2+49x+59$
- $y^2=17x^6+102x^5+120x^4+122x^3+74x^2+38x+52$
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.as 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.