Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 10 x + 53 x^{2} - 190 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.114247126664$, $\pm0.432392726124$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1089600.1 |
Galois group: | $D_{4}$ |
Jacobians: | $6$ |
Isomorphism classes: | 12 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $215$ | $132225$ | $47189060$ | $16909065225$ | $6125993285375$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $368$ | $6880$ | $129748$ | $2474050$ | $47057438$ | $893979670$ | $16983847588$ | $322687758640$ | $6131068120448$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=8x^6+13x^5+18x^4+6x^3+7x+18$
- $y^2=3x^6+x^5+17x^4+13x^3+x^2+2x+3$
- $y^2=18x^6+18x^5+14x^4+13x^3+12x^2+17x+2$
- $y^2=2x^6+5x^5+13x^4+6x^3+x^2+11x+18$
- $y^2=14x^6+3x^5+14x^4+11x^3+18x^2+3x+17$
- $y^2=15x^6+9x^5+15x^4+12x^3+8x^2+5x+10$
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 4.0.1089600.1. |
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.k_cb | $2$ | (not in LMFDB) |