Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 8 x + 35 x^{2} - 152 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.0916535161451$, $\pm0.513108055291$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.2685840.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $237$ | $132009$ | $46183716$ | $16852928985$ | $6131645400477$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $12$ | $368$ | $6732$ | $129316$ | $2476332$ | $47061326$ | $893880468$ | $16983490756$ | $322689077748$ | $6131075578928$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=11x^6+10x^5+3x^4+5x^3+7x^2+15x+3$
- $y^2=3x^6+17x^5+16x^4+9x^3+6x^2+12x+7$
- $y^2=6x^6+2x^5+13x^4+9x^3+16x^2+11x+2$
- $y^2=17x^6+10x^5+8x^4+12x^3+6x^2+3x+14$
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 4.0.2685840.1. |
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.i_bj | $2$ | (not in LMFDB) |