Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + x - 8 x^{2} + 13 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.233764680725$, $\pm0.843070151877$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-47 +4 \sqrt{137}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $10$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $176$ | $26048$ | $4972352$ | $829055744$ | $137895736176$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $15$ | $153$ | $2262$ | $29025$ | $371395$ | $4832454$ | $62726847$ | $815717985$ | $10604248302$ | $137857968913$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=7 x^6+6 x^5+9 x^4+9 x^3+x^2+3 x+3$
- $y^2=2 x^6+9 x^5+2 x^4+7 x^3+3 x^2+11 x+9$
- $y^2=5 x^6+5 x^5+x^4+5 x^3+8 x^2+12 x+12$
- $y^2=x^6+9 x^5+12 x^4+3 x^3+9 x^2+8 x+12$
- $y^2=9 x^5+11 x^4+10 x^3+5 x^2+7 x+4$
- $y^2=12 x^6+8 x^5+8 x^4+4 x^3+12 x^2+6 x+12$
- $y^2=10 x^6+11 x^5+6 x^4+2 x^3+6 x^2+6 x+11$
- $y^2=6 x^5+7 x^4+x^3+3 x^2+7 x$
- $y^2=3 x^6+3 x^5+10 x^4+4 x^3+5 x^2+2$
- $y^2=x^6+3 x^5+9 x^4+4 x^3+5 x^2+4 x+4$
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 \(\Q(\sqrt{-47 +4 \sqrt{137}})\). |
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.ab_ai | $2$ | 2.169.ar_om |