Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 11 x + 63 x^{2} - 209 x^{3} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.148084750558$, $\pm0.380020549717$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.443205.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 8 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $205$ | $132225$ | $47885335$ | $17004796125$ | $6128994732400$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $9$ | $367$ | $6981$ | $130483$ | $2475264$ | $47050027$ | $893959971$ | $16984039603$ | $322688549499$ | $6131062547302$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=13x^6+17x^5+5x^4+12x^3+18x^2+7x+14$
- $y^2=2x^6+4x^5+14x^4+11x^3+12x^2+16$
- $y^2=3x^6+6x^5+18x^4+x^3+12x+12$
- $y^2=8x^6+16x^5+9x^4+8x^3+18x^2+8x+12$
- $y^2=12x^6+15x^5+2x^4+5x^2+7x+18$
- $y^2=15x^6+16x^5+5x^4+9x^3+7x^2+7x+18$
- $y^2=18x^6+11x^5+x^4+8x^3+13x+8$
- $y^2=7x^6+16x^5+6x^4+3x^3+2x^2+3x+3$
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 4.0.443205.2. |
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.l_cl | $2$ | (not in LMFDB) |