Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 10 x + 55 x^{2} - 190 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.145032844023$, $\pm0.419866368263$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.67136.2 |
Galois group: | $D_{4}$ |
Jacobians: | $9$ |
Isomorphism classes: | 9 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $217$ | $133889$ | $47604592$ | $16956907961$ | $6128712789577$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $372$ | $6940$ | $130116$ | $2475150$ | $47059302$ | $893990170$ | $16983903108$ | $322687477540$ | $6131063008852$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 9 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=4x^6+7x^5+14x^4+4x^3+15x^2+8x+13$
- $y^2=13x^6+5x^5+16x^4+11x^3+15x^2+14x+10$
- $y^2=2x^6+13x^5+4x^4+18x^3+2x^2+9x+12$
- $y^2=14x^6+13x^4+11x^3+8x^2+15x+3$
- $y^2=4x^6+17x^5+17x^3+7x^2+16x+14$
- $y^2=14x^6+x^5+16x^4+11x^3+8x^2+14x+3$
- $y^2=16x^6+13x^5+7x^4+10x^3+9x^2+18x+2$
- $y^2=12x^6+5x^5+3x^4+5x^3+8x^2+3x+12$
- $y^2=10x^6+18x^5+9x^4+8x^3+10x^2+6x+9$
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 4.0.67136.2. |
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_cd | $2$ | (not in LMFDB) |