Invariants
Base field: | $\F_{17}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 6 x + 17 x^{2} )( 1 - 4 x + 17 x^{2} )$ |
$1 - 10 x + 58 x^{2} - 170 x^{3} + 289 x^{4}$ | |
Frobenius angles: | $\pm0.240632536990$, $\pm0.338793663197$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $8$ |
Isomorphism classes: | 24 |
This isogeny class is not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $168$ | $88704$ | $25290216$ | $7045226496$ | $2016775051368$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $8$ | $306$ | $5144$ | $84350$ | $1420408$ | $24129522$ | $410300584$ | $6975697534$ | $118587959528$ | $2015994055986$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=5x^6+4x^5+12x^4+14x^2+13x+14$
- $y^2=7x^6+16x^5+6x^4+6x^3+6x^2+16x+7$
- $y^2=14x^6+9x^5+15x^4+x^3+15x^2+9x+14$
- $y^2=5x^6+9x^4+4x^3+9x^2+5$
- $y^2=11x^6+5x^5+10x^4+7x^3+12x^2+14x+5$
- $y^2=6x^6+7x^5+6x^4+15x^3+3x^2+6x+5$
- $y^2=x^6+10x^5+14x^4+8x^3+3x^2+10x+16$
- $y^2=2x^5+7x^4+15x^3+7x^2+2x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17}$.
Endomorphism algebra over $\F_{17}$The isogeny class factors as 1.17.ag $\times$ 1.17.ae and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
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.ac_k | $2$ | (not in LMFDB) |
2.17.c_k | $2$ | (not in LMFDB) |
2.17.k_cg | $2$ | (not in LMFDB) |