Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 4 x + 22 x^{2} + 52 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.463350950174$, $\pm0.733526888750$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-5 + \sqrt{2}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $14$ |
| Isomorphism classes: | 22 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $248$ | $33728$ | $4732088$ | $816487424$ | $137385869688$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $198$ | $2154$ | $28590$ | $370018$ | $4828854$ | $62774394$ | $815643870$ | $10604386866$ | $137859161638$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):
- $y^2=12 x^5+10 x^4+5 x^3+6 x^2+9 x+10$
- $y^2=x^6+9 x^4+x^3+12 x^2+10 x+11$
- $y^2=9 x^5+3 x^4+2 x^3+3 x^2+12 x+1$
- $y^2=x^6+4 x^5+12 x^4+2 x^3+10 x^2+4 x+4$
- $y^2=11 x^6+12 x^5+7 x^4+10 x^3+6 x^2+12 x+11$
- $y^2=10 x^6+3 x^5+2 x^4+12 x^3+5 x^2+4 x+1$
- $y^2=11 x^6+11 x^5+9 x^4+x^3+5 x^2+11 x+1$
- $y^2=12 x^6+11 x^5+4 x^4+11 x^3+2 x^2+7 x+2$
- $y^2=11 x^6+7 x^5+11 x^4+3 x^3+3 x^2+10 x+9$
- $y^2=10 x^6+4 x^5+5 x^4+3 x^3+7 x^2+x+9$
- $y^2=11 x^6+3 x^4+11 x^3+6 x^2+11 x$
- $y^2=10 x^6+5 x^5+5 x^4+7 x^3+8 x^2+8 x+9$
- $y^2=4 x^6+8 x^5+8 x^4+4 x^3+12 x^2+x+10$
- $y^2=10 x^6+3 x^5+8 x^3+6 x^2+11 x+3$
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 \(\Q(\sqrt{-5 + \sqrt{2}})\). |
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.ae_w | $2$ | 2.169.bc_pq |