Invariants
| Base field: | $\F_{139}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 21 x + 139 x^{2} )^{2}$ |
| $1 - 42 x + 719 x^{2} - 5838 x^{3} + 19321 x^{4}$ | |
| Frobenius angles: | $\pm0.150285916016$, $\pm0.150285916016$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $11$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $14161$ | $367067281$ | $7209847933456$ | $139362681831003225$ | $2692485785724163187521$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $98$ | $18996$ | $2684612$ | $373325188$ | $51889491878$ | $7212559647606$ | $1002544493394962$ | $139353668413373188$ | $19370159750289295388$ | $2692452204195076097556$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 11 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=10x^6+39x^5+27x^4+71x^3+113x^2+66x+55$
- $y^2=88x^6+86x^5+74x^4+31x^3+125x^2+3x+68$
- $y^2=14x^6+33x^5+33x^4+47x^3+33x^2+33x+14$
- $y^2=135x^6+38x^5+14x^4+71x^3+x^2+93x+56$
- $y^2=132x^6+136x^5+33x^4+47x^3+33x^2+136x+132$
- $y^2=128x^6+111x^5+22x^4+77x^3+106x^2+77x+62$
- $y^2=132x^6+81x^5+45x^4+42x^3+45x^2+81x+132$
- $y^2=56x^6+131x^5+110x^4+72x^3+110x^2+131x+56$
- $y^2=23x^6+96x^5+89x^4+44x^3+6x^2+25x+122$
- $y^2=116x^6+5x^5+122x^4+21x^3+122x^2+5x+116$
- $y^2=53x^6+101x^5+127x^4+83x^3+122x^2+45x+135$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{139}$.
Endomorphism algebra over $\F_{139}$| The isogeny class factors as 1.139.av 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-115}) \)$)$ |
Base change
This is a primitive isogeny class.