Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 11 x + 64 x^{2} - 209 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.165808925970$, $\pm0.370946582130$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.349112.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $206$ | $133076$ | $48115832$ | $17034792608$ | $6130763962866$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $9$ | $369$ | $7014$ | $130713$ | $2475979$ | $47049186$ | $893941953$ | $16983957489$ | $322688277282$ | $6131060869489$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=14x^6+17x^5+8x^4+9x^3+9x^2+16x+14$
- $y^2=18x^6+10x^5+10x^4+x^3+15x^2+15x+2$
- $y^2=3x^6+17x^5+7x^4+9x^3+11x^2+11x+18$
- $y^2=x^6+2x^5+5x^4+9x^3+10x^2+11x+14$
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 4.0.349112.1. |
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.l_cm | $2$ | (not in LMFDB) |