Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 6 x + 21 x^{2} - 78 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.115489212201$, $\pm0.532795966182$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.1056832.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 6 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $107$ | $29425$ | $4674188$ | $805803625$ | $138191347547$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $176$ | $2126$ | $28212$ | $372188$ | $4832894$ | $62751116$ | $815747428$ | $10604868518$ | $137859530336$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=7x^6+2x^5+2x^4+6x^3+9x^2+5x+1$
- $y^2=9x^6+11x^5+3x^4+4x^3+8x^2+x+5$
- $y^2=2x^6+10x^5+2x^4+5x^3+5x^2+10x+10$
- $y^2=12x^6+2x^5+12x^4+7x^3+9x^2+9x+8$
- $y^2=7x^6+4x^5+6x^3+6x^2+12x+12$
- $y^2=11x^6+7x^5+11x^4+2x^3+11x^2+10x+7$
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 4.0.1056832.1. |
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_v | $2$ | 2.169.g_agb |