Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 9 x + 41 x^{2} - 171 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.0387402455333$, $\pm0.487338161656$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.404685.1 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
Isomorphism classes: | 2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $223$ | $130009$ | $46130449$ | $16808213565$ | $6123139391248$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $11$ | $363$ | $6725$ | $128971$ | $2472896$ | $47048991$ | $893848799$ | $16983168931$ | $322686779645$ | $6131069107878$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+7x^5+12x^4+11x^3+9x^2+13x+14$
- $y^2=2x^6+7x^5+9x^4+7x^3+2x^2+14x+15$
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 4.0.404685.1. |
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.j_bp | $2$ | (not in LMFDB) |