Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 8 x + 37 x^{2} - 104 x^{3} + 169 x^{4}$ |
Frobenius angles: | $\pm0.167453355204$, $\pm0.421338968608$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.16025.1 |
Galois group: | $D_{4}$ |
Jacobians: | $10$ |
Isomorphism classes: | 10 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $95$ | $30305$ | $4971920$ | $815356025$ | $137825337375$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $180$ | $2262$ | $28548$ | $371206$ | $4831590$ | $62777742$ | $815780868$ | $10604331246$ | $137857522900$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=7x^6+11x^5+10x^4+2x^3+10x^2+7x+3$
- $y^2=6x^6+7x^5+x^2+4x+6$
- $y^2=5x^6+9x^5+8x^4+10x^2+x+11$
- $y^2=2x^6+2x^5+8x^4+6x^2+2x+8$
- $y^2=10x^6+6x^5+2x^4+12x^3+8x^2+9x+12$
- $y^2=x^6+8x^5+12x^4+9x^3+9x^2+10x+11$
- $y^2=2x^6+5x^5+4x^4+5x^3+12x^2+5x+2$
- $y^2=11x^6+2x^5+5x^3+x^2+9x+6$
- $y^2=5x^6+4x^5+6x^4+12x^3+8x^2+12$
- $y^2=7x^6+2x^5+11x^4+3x^3+2x^2+12x+7$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13}$.
Endomorphism algebra over $\F_{13}$The endomorphism algebra of this simple isogeny class is 4.0.16025.1. |
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 |
---|---|---|
2.13.i_bl | $2$ | 2.169.k_br |