Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 2 x + 3 x^{2} - 26 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.195060026239$, $\pm0.681837234905$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-27 +4 \sqrt{6}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $12$ |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $145$ | $29145$ | $4679440$ | $829612425$ | $138438127225$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $172$ | $2130$ | $29044$ | $372852$ | $4826374$ | $62766996$ | $815728036$ | $10604163930$ | $137858383132$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=6 x^6+6 x^5+4 x^4+12 x^3+11 x^2+11 x+7$
- $y^2=9 x^6+x^5+10 x^4+5 x^3+9 x^2+4 x+4$
- $y^2=10 x^6+8 x^5+9 x^4+5 x^3+9 x^2+4 x+8$
- $y^2=2 x^6+5 x^5+2 x^4+2 x^3+10 x^2+9$
- $y^2=11 x^6+10 x^5+x^4+3 x^3+2 x^2+x+4$
- $y^2=6 x^6+12 x^4+6 x^3+7 x^2+11 x+4$
- $y^2=11 x^6+8 x^5+9 x^4+11 x^3+6 x^2+12 x+9$
- $y^2=9 x^6+3 x^5+5 x^4+4 x^3+12 x^2+12 x+8$
- $y^2=x^6+5 x^5+6 x^4+9 x^3+10 x^2+10 x+8$
- $y^2=12 x^6+8 x^5+4 x^4+2 x^3+12 x^2+5 x+6$
- $y^2=7 x^6+5 x^5+12 x^4+5 x^3+8 x^2+8 x+10$
- $y^2=2 x^6+7 x^5+7 x^4+3 x^3+6 x^2+12 x+4$
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 \(\Q(\sqrt{-27 +4 \sqrt{6}})\). |
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.c_d | $2$ | 2.169.c_jj |