Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 8 x + 53 x^{2} + 208 x^{3} + 901 x^{4} + 2312 x^{5} + 4913 x^{6}$ |
| Frobenius angles: | $\pm0.406491736863$, $\pm0.696720321556$, $\pm0.738142595788$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.33165747200.1 |
| Galois group: | $S_4\times C_2$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $8396$ | $28042640$ | $115320546092$ | $588240363929600$ | $2855191516265647276$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $26$ | $332$ | $4778$ | $84324$ | $1416266$ | $24125228$ | $410447098$ | $6975676796$ | $118587392666$ | $2015994977772$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 19 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^8+15 x^7+4 x^6+11 x^5+x^4+9 x^3+16 x^2+16 x+8$
- $y^2=x^8+2 x^7+11 x^6+5 x^5+9 x^3+13 x^2+7 x+11$
- $y^2=3 x^8+14 x^7+11 x^6+11 x^4+x^3+14 x^2+7 x$
- $y^2=x^8+8 x^7+8 x^6+2 x^5+10 x^4+16 x^3+x^2+4 x+16$
- $y^2=x^8+9 x^7+5 x^6+x^5+7 x^4+14 x^3+10 x^2+14 x+5$
- $y^2=3 x^8+10 x^7+8 x^6+14 x^5+5 x^4+13 x^3+4 x^2+2 x$
- $y^2=x^8+14 x^7+5 x^6+7 x^5+12 x^4+13 x^3+9 x^2+8 x$
- $y^2=x^8+5 x^7+15 x^6+11 x^5+2 x^4+5 x^3+11 x^2+3 x+16$
- $y^2=x^8+6 x^7+2 x^6+11 x^5+12 x^4+6 x^3+2 x^2+11 x+12$
- $y^2=x^8+x^7+2 x^6+11 x^5+14 x^4+13 x^3+12 x^2+9 x+1$
- $y^2=x^8+4 x^7+11 x^6+2 x^5+x^4+11 x^2+16 x+11$
- $y^2=x^8+x^7+15 x^6+6 x^5+15 x^4+11 x^3+2 x^2+x+2$
- $y^2=x^8+6 x^7+3 x^6+13 x^5+2 x^4+7 x^3+9 x^2+6 x+9$
- $y^2=3 x^8+12 x^7+9 x^4+3 x^3+10 x^2+14 x+15$
- $y^2=x^8+13 x^7+3 x^6+13 x^5+5 x^4+4 x^3+16 x+2$
- $y^2=3 x^8+14 x^7+15 x^6+11 x^5+3 x^4+8 x^3+10 x+9$
- $y^2=x^8+11 x^6+9 x^5+12 x^4+10 x^3+5 x^2+2 x+15$
- $y^2=x^8+12 x^7+13 x^5+12 x^3+13 x^2+16 x+2$
- $y^2=x^8+15 x^7+15 x^6+8 x^5+7 x^3+4 x^2+9 x+7$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17}$.
Endomorphism algebra over $\F_{17}$| The endomorphism algebra of this simple isogeny class is 6.0.33165747200.1. |
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 |
|---|---|---|
| 3.17.ai_cb_aia | $2$ | (not in LMFDB) |