Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - x + 16 x^{2} - 17 x^{3} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.303564039335$, $\pm0.651227252100$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-15 + \sqrt{73}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $28$ |
| Isomorphism classes: | 28 |
| 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: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $288$ | $93312$ | $24109056$ | $7029379584$ | $2018153284128$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $17$ | $321$ | $4910$ | $84161$ | $1421377$ | $24119838$ | $410308657$ | $6975852289$ | $118587711182$ | $2015996831841$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):
- $y^2=4 x^6+11 x^5+8 x^4+13 x^3+9 x^2+x+1$
- $y^2=10 x^6+10 x^5+15 x^4+13 x^3+15 x^2+2 x+14$
- $y^2=7 x^6+13 x^5+15 x^4+x^3+3 x^2+16 x+8$
- $y^2=11 x^6+5 x^5+4 x^4+16 x^3+4 x+11$
- $y^2=15 x^6+8 x^5+10 x^4+13 x^3+16 x^2+10 x+7$
- $y^2=2 x^6+x^5+8 x^4+16 x^3+x^2+6 x$
- $y^2=11 x^6+3 x^4+4 x^3+12 x^2+x+1$
- $y^2=15 x^6+5 x^5+12 x^4+13 x^3+11 x^2+6 x+6$
- $y^2=9 x^6+14 x^5+16 x^4+7 x^3+9 x^2+8 x$
- $y^2=13 x^6+10 x^5+10 x^4+15 x^3+13 x+7$
- $y^2=7 x^6+2 x^5+11 x^4+15 x^3+16 x+3$
- $y^2=12 x^6+11 x^5+4 x^4+5 x^3+7 x+7$
- $y^2=2 x^6+14 x^5+9 x^4+13 x^2+2 x+6$
- $y^2=16 x^6+15 x^4+3 x^3+16 x^2+10 x+12$
- $y^2=12 x^6+11 x^5+16 x^4+16 x^3+13 x^2+16 x+16$
- $y^2=11 x^6+7 x^5+5 x^4+9 x^3+16 x^2+5 x+12$
- $y^2=4 x^5+10 x^4+9 x^3+11 x^2+3 x+14$
- $y^2=x^6+12 x^5+6 x^4+11 x^3+2 x^2+12 x+6$
- $y^2=2 x^6+3 x^5+2 x^4+x^3+3 x^2+4 x+10$
- $y^2=13 x^6+3 x^5+5 x^4+3 x^3+13 x^2+12 x+8$
- $y^2=13 x^6+11 x^5+15 x^4+16 x^3+11 x^2+11 x+8$
- $y^2=9 x^6+4 x^5+7 x^3+16 x^2+10 x+3$
- $y^2=15 x^6+7 x^5+9 x^4+5 x^3+9 x^2+14 x+3$
- $y^2=7 x^6+13 x^5+8 x^4+3 x^3+5 x^2+11 x+13$
- $y^2=5 x^6+9 x^5+7 x^4+4 x^3+3 x^2+12 x+15$
- $y^2=8 x^5+7 x^4+x^3+16 x^2+3 x+9$
- $y^2=13 x^6+16 x^5+10 x^4+12 x^3+12 x^2+13 x+16$
- $y^2=4 x^6+16 x^5+x^4+15 x^3+4 x^2+x+6$
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 \(\Q(\sqrt{-15 + \sqrt{73}})\). |
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 |
|---|---|---|
| 2.17.b_q | $2$ | (not in LMFDB) |