Invariants
| Base field: | $\F_{139}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 42 x + 717 x^{2} - 5838 x^{3} + 19321 x^{4}$ |
| Frobenius angles: | $\pm0.100495437984$, $\pm0.187984873465$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.591424.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $7$ |
This isogeny class is simple and geometrically 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $14159$ | $366987121$ | $7209169651172$ | $139359557580163561$ | $2692475608143040452599$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $98$ | $18992$ | $2684360$ | $373316820$ | $51889295738$ | $7212556063982$ | $1002544440825998$ | $139353667812016804$ | $19370159745842629640$ | $2692452204204233629952$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=123x^6+94x^5+51x^4+104x^3+57x^2+60x+89$
- $y^2=115x^6+98x^5+97x^4+71x^3+77x^2+36x+72$
- $y^2=113x^6+23x^5+10x^4+10x^3+8x^2+76x+122$
- $y^2=63x^6+3x^5+74x^4+54x^3+9x^2+83x+76$
- $y^2=82x^6+60x^5+87x^4+127x^3+90x^2+61x+64$
- $y^2=85x^6+112x^5+x^4+14x^3+40x^2+40x+101$
- $y^2=68x^6+98x^5+x^4+20x^3+108x^2+107x+51$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{139}$.
Endomorphism algebra over $\F_{139}$| The endomorphism algebra of this simple isogeny class is 4.0.591424.1. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.139.bq_bbp | $2$ | (not in LMFDB) |