Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 5 x + 40 x^{2} - 116 x^{3} + 520 x^{4} - 845 x^{5} + 2197 x^{6}$ |
| Frobenius angles: | $\pm0.327534793960$, $\pm0.378603702997$, $\pm0.562313797224$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.1176131467.1 |
| Galois group: | $S_4\times C_2$ |
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: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1792$ | $6673408$ | $11229261568$ | $23246229508096$ | $51096293079580672$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $9$ | $225$ | $2325$ | $28497$ | $370644$ | $4821249$ | $62730593$ | $815808417$ | $10604963829$ | $137858112300$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=2 x^8+9 x^7+8 x^6+10 x^5+6 x^4+2 x^3+10 x^2+3 x+11$
- $y^2=2 x^8+4 x^7+12 x^6+7 x^5+6 x^4+11 x^3+8 x^2+9 x+6$
- $y^2=x^8+3 x^7+5 x^6+6 x^4+9 x^3+6 x^2+11 x+11$
- $y^2=2 x^8+x^7+3 x^6+11 x^5+12 x^4+x^3+8 x^2+7 x+5$
- $y^2=x^8+8 x^7+2 x^6+6 x^5+12 x^4+4 x^3+9 x^2+5$
- $y^2=x^8+7 x^7+3 x^6+4 x^5+2 x^4+10 x^3+7 x^2+12 x+4$
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 6.0.1176131467.1. |
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 |
|---|---|---|
| 3.13.f_bo_em | $2$ | (not in LMFDB) |