Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 6 x + 20 x^{2} - 78 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.0978612922159$, $\pm0.538629668498$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.878400.4 |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 8 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $106$ | $29044$ | $4635274$ | $804054096$ | $138102032266$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $174$ | $2108$ | $28150$ | $371948$ | $4831278$ | $62745320$ | $815742814$ | $10604873864$ | $137859492414$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=8x^6+x^5+x^4+6x^3+7x^2+11x+3$
- $y^2=3x^6+3x^5+12x^4+8x^3+4x^2+x+2$
- $y^2=8x^6+11x^5+7x^4+9x^3+5x^2+7$
- $y^2=2x^6+10x^5+12x^4+2x^3+5x^2+12x+2$
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 4.0.878400.4. |
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.g_u | $2$ | 2.169.e_ahq |