Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 4 x + 12 x^{2} - 52 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.166874660148$, $\pm0.600663352317$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-30 +12 \sqrt{2}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $16$ |
| Isomorphism classes: | 20 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $126$ | $29988$ | $4664142$ | $817712784$ | $138705588126$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $178$ | $2122$ | $28630$ | $373570$ | $4829794$ | $62749522$ | $815805790$ | $10604558122$ | $137857373938$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):
- $y^2=5 x^6+6 x^5+5 x^4+6 x^3+3 x^2+10 x+9$
- $y^2=2 x^6+11 x^5+8 x^4+7 x^3+11 x^2+2$
- $y^2=8 x^6+7 x^5+5 x^4+10 x^3+6 x^2+4 x+10$
- $y^2=10 x^6+12 x^5+10 x^3+5 x^2+4 x+4$
- $y^2=3 x^6+5 x^5+4 x^4+10 x^3+9 x^2+6 x+11$
- $y^2=12 x^5+2 x^4+4 x^3+11 x^2+2 x+6$
- $y^2=5 x^6+5 x^4+9 x^3+x^2+6 x+6$
- $y^2=9 x^5+11 x^3+12 x+4$
- $y^2=6 x^6+6 x^5+2 x^4+4 x^3+4 x^2+6$
- $y^2=11 x^6+2 x^5+3 x^4+3 x^3+11 x^2+7 x+2$
- $y^2=5 x^6+8 x^5+4 x^4+3 x^3+6 x^2+11 x+5$
- $y^2=2 x^6+6 x^5+3 x^4+2 x^3+4 x^2+12 x+3$
- $y^2=2 x^6+10 x^5+11 x^4+10 x^3+4 x^2+4 x+4$
- $y^2=9 x^6+7 x^5+5 x^4+x^3+6 x^2+3 x+2$
- $y^2=8 x^6+8 x^5+12 x^4+9 x^3+2 x^2+5 x+2$
- $y^2=8 x^6+2 x^5+5 x^4+2 x^3+4 x^2+12 x+4$
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 \(\Q(\sqrt{-30 +12 \sqrt{2}})\). |
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_m | $2$ | 2.169.i_co |