Invariants
Base field: | $\F_{17}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 12 x + 68 x^{2} - 204 x^{3} + 289 x^{4}$ |
Frobenius angles: | $\pm0.144218155731$, $\pm0.312293972145$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.39168.3 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $142$ | $81508$ | $24674062$ | $7021751184$ | $2017512450862$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $282$ | $5022$ | $84070$ | $1420926$ | $24137466$ | $410345382$ | $6975904510$ | $118588855302$ | $2015997043962$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=12x^6+13x^5+2x^4+4x^3+8x^2+10x+16$
- $y^2=3x^6+8x^5+3x^4+13x^3+10x^2+4x+3$
- $y^2=5x^6+2x^5+3x^4+11x^3+4x^2+14x+5$
- $y^2=3x^6+14x^5+2x^4+15x^3+13x^2+15x+16$
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 4.0.39168.3. |
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.m_cq | $2$ | (not in LMFDB) |