Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 9 x + 44 x^{2} - 171 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.101896915851$, $\pm0.473494688146$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1845432.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $226$ | $132436$ | $46685272$ | $16868108448$ | $6128151768766$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $11$ | $369$ | $6806$ | $129433$ | $2474921$ | $47062626$ | $893941787$ | $16983576049$ | $322688021618$ | $6131074533489$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=13x^5+3x^4+14x^3+5x^2+8x+10$
- $y^2=11x^6+16x^5+10x^4+11x^3+x^2+10x+12$
- $y^2=15x^6+x^5+7x^4+16x^2+2x+15$
- $y^2=17x^6+13x^5+13x^4+6x^3+5x^2+5$
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 4.0.1845432.1. |
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_bs | $2$ | (not in LMFDB) |