Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 6 x + 22 x^{2} - 78 x^{3} + 169 x^{4}$ |
Frobenius angles: | $\pm0.131379257950$, $\pm0.526761541651$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.8112.1 |
Galois group: | $D_{4}$ |
Jacobians: | $18$ |
Isomorphism classes: | 24 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $108$ | $29808$ | $4713228$ | $807439104$ | $138258290988$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $8$ | $178$ | $2144$ | $28270$ | $372368$ | $4834114$ | $62755064$ | $815742430$ | $10604813816$ | $137859397138$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=7x^6+7x^5+10x^4+11x^3+7x^2+9x+11$
- $y^2=5x^6+12x^5+5x^4+3x^3+12x^2+11x+8$
- $y^2=2x^6+7x^4+x^3+7x^2+x$
- $y^2=6x^6+9x^5+9x^4+5x^3+9x^2+9x+7$
- $y^2=8x^6+8x^5+10x^3+12x^2+x+7$
- $y^2=11x^6+9x^5+7x^3+8x^2+11$
- $y^2=11x^5+7x^3+12x^2+9x+6$
- $y^2=x^6+12x^5+10x^4+5x^3+x^2+8x+8$
- $y^2=10x^5+7x^4+10x^3+7x+9$
- $y^2=3x^6+x^5+x^4+3x^3+12x^2+6x+1$
- $y^2=8x^6+7x^5+12x^4+7x^3+8x^2+7x+5$
- $y^2=6x^6+9x^5+8x^4+5x^3+4x^2+2x+6$
- $y^2=6x^6+9x^5+5x^4+3x^2+5x+11$
- $y^2=7x^6+7x^4+11x^3+11x^2+3x+8$
- $y^2=7x^6+5x^5+8x^4+11x^3+4x^2+x+10$
- $y^2=2x^6+9x^5+3x^4+8x^3+11x^2+8x+11$
- $y^2=5x^6+9x^5+3x^4+7x^3+10x^2+9x+4$
- $y^2=11x^6+11x^5+4x^4+3x^3+5x+1$
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 4.0.8112.1. |
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.g_w | $2$ | 2.169.i_aek |