Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 10 x + 61 x^{2} - 190 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.236824788216$, $\pm0.365068605593$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.140864.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $223$ | $138929$ | $48859300$ | $17088405929$ | $6130940329303$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $384$ | $7120$ | $131124$ | $2476050$ | $47037918$ | $893853670$ | $16983569124$ | $322687365760$ | $6131062008304$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+x^5+6x^4+13x^3+16x^2+10x+3$
- $y^2=13x^6+11x^5+16x^4+5x^3+14x^2+16x+11$
- $y^2=13x^6+3x^5+16x^4+9x^3+3x^2+16x+10$
- $y^2=8x^6+8x^5+16x^3+10x^2+9x+10$
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 4.0.140864.1. |
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_cj | $2$ | (not in LMFDB) |