Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - x + 22 x^{2} - 19 x^{3} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.326022808523$, $\pm0.632745285889$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-238 +2 \sqrt{65}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $16$ |
| Isomorphism classes: | 32 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $364$ | $147056$ | $47088496$ | $17047319744$ | $6134754695524$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $19$ | $405$ | $6868$ | $130809$ | $2477589$ | $47021286$ | $893820751$ | $16983905649$ | $322688157292$ | $6131067090325$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):
- $y^2=4 x^5+10 x^4+13 x^3+18 x^2+10 x+11$
- $y^2=16 x^6+18 x^5+10 x^4+4 x^3+17 x^2+6 x+7$
- $y^2=10 x^6+15 x^5+9 x^4+6 x^3+17 x^2+10 x$
- $y^2=2 x^6+7 x^5+18 x^4+2 x^3+16 x^2+6 x+7$
- $y^2=x^6+9 x^5+13 x^4+11 x^3+7 x^2+5 x+7$
- $y^2=3 x^6+4 x^5+15 x^4+5 x^3+5 x^2+11 x+16$
- $y^2=10 x^6+x^5+9 x^4+17 x^3+2 x^2+3$
- $y^2=6 x^6+2 x^5+10 x^4+4 x^3+18 x^2+12 x+11$
- $y^2=12 x^6+9 x^5+15 x^4+16 x^3+9 x^2+9 x+6$
- $y^2=9 x^6+11 x^5+x^4+x^3+10 x^2+11 x+18$
- $y^2=11 x^6+3 x^5+2 x^4+11 x^3+5 x^2+7 x+12$
- $y^2=2 x^6+10 x^5+11 x^4+11 x^3+4 x^2+13 x$
- $y^2=10 x^6+16 x^5+12 x^4+12 x^3+10 x^2+5 x+6$
- $y^2=4 x^6+9 x^5+7 x^4+10 x^3+9 x^2+x$
- $y^2=18 x^6+7 x^5+8 x^4+9 x^3+13 x^2+3 x+17$
- $y^2=5 x^6+18 x^4+11 x^3+10 x^2+9 x+14$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19}$.
Endomorphism algebra over $\F_{19}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-238 +2 \sqrt{65}})\). |
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 |
|---|---|---|
| 2.19.b_w | $2$ | (not in LMFDB) |