Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 8 x + 39 x^{2} - 104 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.207527242884$, $\pm0.398160485086$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.129168.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 4 |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $1$ |
| Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $97$ | $31137$ | $5080084$ | $821300649$ | $137854789897$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $184$ | $2310$ | $28756$ | $371286$ | $4828462$ | $62763294$ | $815750884$ | $10604257566$ | $137857028584$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=5x^6+9x^5+10x^4+7x^3+4x^2+3x+8$
- $y^2=11x^6+11x^5+8x^4+9x^3+4x^2+8x+7$
- $y^2=8x^6+3x^5+11x^4+5x^3+9x^2+7x+2$
- $y^2=6x^6+12x^5+9x^4+5x^3+6x^2+x+8$
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 4.0.129168.1. |
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.i_bn | $2$ | 2.169.o_hn |