Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 10 x + 60 x^{2} - 190 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.219146719950$, $\pm0.377690954553$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.288576.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $222$ | $138084$ | $48649302$ | $17067734736$ | $6131186130102$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $382$ | $7090$ | $130966$ | $2476150$ | $47044222$ | $893891470$ | $16983636958$ | $322686919690$ | $6131058389902$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=12x^6+11x^5+10x^4+12x^3+12x^2+4x+15$
- $y^2=16x^6+8x^5+17x^4+12x^3+13x^2+6x+12$
- $y^2=2x^6+14x^5+6x^4+x^3+8x^2+5x$
- $y^2=2x^6+8x^5+x^4+5x^3+8x^2+2x+13$
- $y^2=14x^6+16x^5+14x^3+16x^2+15x+11$
- $y^2=8x^6+13x^5+11x^4+11x^3+15x^2+17x$
- $y^2=14x^6+5x^5+13x^4+2x^3+7x+16$
- $y^2=3x^6+9x^5+4x^4+6x^3+3x^2+14$
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 4.0.288576.1. |
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_ci | $2$ | (not in LMFDB) |