Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 10 x + 57 x^{2} - 190 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.173854422602$, $\pm0.405491683409$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.820800.3 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 8 |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $219$ | $135561$ | $48021444$ | $17002738425$ | $6130443137979$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $376$ | $7000$ | $130468$ | $2475850$ | $47056606$ | $893970430$ | $16983838468$ | $322686954520$ | $6131058005656$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=12x^6+3x^5+13x^4+6x^3+5x^2+8x+15$
- $y^2=12x^6+8x^5+2x^4+x^3+2x^2+6x+14$
- $y^2=18x^6+4x^5+12x^4+6x^3+6x^2+6x+8$
- $y^2=17x^6+16x^5+16x^4+9x^3+3x^2+12x+11$
- $y^2=3x^6+x^5+13x^4+13x^3+12x^2+11x+18$
- $y^2=14x^6+9x^5+4x^3+2x^2+11x+10$
- $y^2=15x^6+2x^5+3x^4+17x^3+5x^2+10$
- $y^2=12x^6+17x^5+12x^4+17x^3+3x^2+15x+17$
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 4.0.820800.3. |
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.k_cf | $2$ | (not in LMFDB) |