Invariants
Base field: | $\F_{17}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 10 x + 49 x^{2} - 170 x^{3} + 289 x^{4}$ |
Frobenius angles: | $\pm0.0454542879392$, $\pm0.428461841097$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.142400.3 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $159$ | $82521$ | $23939676$ | $6905439801$ | $2010392171439$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $8$ | $288$ | $4874$ | $82676$ | $1415908$ | $24133302$ | $410357284$ | $6975725668$ | $118587062858$ | $2015991718128$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=14x^6+15x^5+5x^4+6x^3+10x^2+5x+3$
- $y^2=14x^6+15x^5+5x^4+11x^3+13x^2+10x+7$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17}$.
Endomorphism algebra over $\F_{17}$The endomorphism algebra of this simple isogeny class is 4.0.142400.3. |
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 |
---|---|---|
2.17.k_bx | $2$ | (not in LMFDB) |