Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 4 x + 10 x^{2} - 52 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.145364093642$, $\pm0.611383306531$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-22 +2 \sqrt{5}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $14$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $124$ | $29264$ | $4612924$ | $816582656$ | $138601292764$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $174$ | $2098$ | $28590$ | $373290$ | $4828638$ | $62754226$ | $815834334$ | $10604614474$ | $137857904014$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):
- $y^2=4 x^6+9 x^5+7 x^4+10 x^3+8 x^2+5 x+4$
- $y^2=2 x^6+2 x^5+11 x^2+2 x+7$
- $y^2=12 x^6+10 x^5+5 x^4+6 x^3+3 x^2+11 x$
- $y^2=6 x^6+3 x^5+12 x^4+5 x^3+7 x^2+11 x+1$
- $y^2=2 x^6+7 x^5+x^4+3 x^3+x^2+5 x+4$
- $y^2=12 x^6+3 x^4+6 x^3+3 x^2+8$
- $y^2=2 x^6+2 x^5+10 x^4+9 x^3+5 x+12$
- $y^2=5 x^6+8 x^5+6 x^4+4 x^3+5 x^2+2 x+2$
- $y^2=2 x^6+11 x^5+10 x^4+10 x^3+12 x^2+11 x+6$
- $y^2=2 x^6+5 x^4+9 x^3+4 x^2+7 x+7$
- $y^2=6 x^6+3 x^5+5 x^4+11 x^3+4 x^2+6 x+4$
- $y^2=9 x^6+8 x^4+2 x^3+4 x^2+5 x$
- $y^2=5 x^6+9 x^5+11 x^4+10 x^3+9 x^2+9 x+10$
- $y^2=4 x^6+3 x^5+5 x^3+7 x^2+11 x+11$
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 \(\Q(\sqrt{-22 +2 \sqrt{5}})\). |
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.e_k | $2$ | 2.169.e_w |