Invariants
| Base field: | $\F_{139}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 23 x + 139 x^{2} )( 1 - 21 x + 139 x^{2} )$ |
| $1 - 44 x + 761 x^{2} - 6116 x^{3} + 19321 x^{4}$ | |
| Frobenius angles: | $\pm0.0707251543800$, $\pm0.150285916016$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $13923$ | $365381289$ | $7204284373104$ | $139349081287597545$ | $2692458503333481860523$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $96$ | $18908$ | $2682540$ | $373288756$ | $51888966096$ | $7212553265846$ | $1002544429932624$ | $139353667965769636$ | $19370159750017368180$ | $2692452204258903929228$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+42x^5+76x^4+93x^3+103x^2+113x+126$
- $y^2=117x^6+90x^5+16x^4+109x^3+9x^2+73x+89$
- $y^2=75x^6+59x^5+41x^4+137x^3+25x^2+110x+74$
- $y^2=136x^6+x^5+135x^4+6x^3+12x^2+9x+81$
- $y^2=132x^6+120x^5+14x^4+121x^3+94x^2+20x+108$
- $y^2=29x^6+5x^5+109x^4+33x^3+110x^2+129x+120$
- $y^2=76x^6+126x^5+75x^4+71x^3+102x^2+32x+95$
- $y^2=66x^6+121x^5+28x^4+17x^3+106x^2+64x+78$
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.av 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.