Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 11 x + 65 x^{2} - 209 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.183908064930$, $\pm0.360590248890$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.26533.1 |
Galois group: | $D_{4}$ |
Jacobians: | $5$ |
Isomorphism classes: | 5 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $207$ | $133929$ | $48346713$ | $17064295677$ | $6132261832272$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $9$ | $371$ | $7047$ | $130939$ | $2476584$ | $47046935$ | $893913309$ | $16983830275$ | $322687980711$ | $6131060307686$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=5x^5+12x^4+15x^3+4x^2+11x+14$
- $y^2=17x^6+17x^5+16x^4+3x^3+9x^2+2x+17$
- $y^2=10x^6+17x^5+2x^4+17x^3+4x+10$
- $y^2=9x^6+x^5+6x^4+6x^3+18x^2+2x+18$
- $y^2=15x^6+x^5+5x^4+8x^3+3x^2+3x+13$
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 4.0.26533.1. |
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_cn | $2$ | (not in LMFDB) |