Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 4 x + 28 x^{2} - 76 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.271143341350$, $\pm0.564023332250$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.9847040.2 |
Galois group: | $D_{4}$ |
Jacobians: | $20$ |
Isomorphism classes: | 20 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $310$ | $145700$ | $47348470$ | $17009018000$ | $6141068412550$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $16$ | $402$ | $6904$ | $130518$ | $2480136$ | $47045442$ | $893755984$ | $16983348318$ | $322688589616$ | $6131067198802$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 20 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=18x^6+12x^4+4x^3+11x^2+5x+16$
- $y^2=18x^6+7x^5+x^4+10x^3+7x^2+14x$
- $y^2=13x^6+6x^5+8x^4+9x^3+4x^2+3$
- $y^2=17x^6+7x^5+13x^4+14x^3+x^2+11x+4$
- $y^2=8x^6+11x^5+18x^4+13x^3+17x^2+11x+6$
- $y^2=18x^6+5x^5+5x^4+9x^3+17x^2+14x+17$
- $y^2=15x^6+14x^5+x^4+11x^3+2x^2+5x+17$
- $y^2=12x^6+18x^5+x^3+4x^2+11x+14$
- $y^2=3x^6+12x^5+10x^4+x^3+12x+2$
- $y^2=5x^6+18x^5+14x^4+15x^3+18x^2+17x+16$
- $y^2=3x^6+14x^5+8x^4+x^3+12x^2+18x+3$
- $y^2=17x^5+7x^4+7x^3+13x^2+18x+16$
- $y^2=8x^6+5x^5+18x^4+4x^3+8x^2+13x+10$
- $y^2=14x^6+14x^5+12x^4+5x^3+5x^2+10x$
- $y^2=13x^6+8x^5+3x^4+16x^2+16x+15$
- $y^2=15x^6+2x^5+11x^4+3x^3+9x^2+11x+14$
- $y^2=14x^6+10x^5+13x^4+18x^3+18x^2+14x+17$
- $y^2=3x^6+15x^5+15x^4+12x^3+4x^2+10x+15$
- $y^2=7x^6+10x^5+5x^4+6x^2+10x+10$
- $y^2=17x^6+13x^5+x^4+12x^3+9x^2+15x$
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 4.0.9847040.2. |
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.e_bc | $2$ | (not in LMFDB) |