Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 10 x + 76 x^{2} - 290 x^{3} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.248744547789$, $\pm0.429855416164$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.4983104.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $28$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $618$ | $752724$ | $604913850$ | $500678884944$ | $420675429278778$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $894$ | $24800$ | $707894$ | $20509600$ | $594836478$ | $17249958340$ | $500245269214$ | $14507133422900$ | $420707207419854$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):
- $y^2=22 x^6+10 x^5+6 x^4+24 x^3+3 x^2+25 x+18$
- $y^2=27 x^6+22 x^5+5 x^4+28 x^3+21 x^2+18 x+3$
- $y^2=2 x^6+x^5+20 x^4+19 x^3+7 x^2+19 x+27$
- $y^2=23 x^6+20 x^5+24 x^4+20 x^3+22 x^2+26$
- $y^2=10 x^6+18 x^5+13 x^4+17 x^3+10 x^2+13 x+21$
- $y^2=23 x^6+x^5+25 x^4+21 x^3+15 x+11$
- $y^2=3 x^6+8 x^5+2 x^4+26 x^3+14 x^2+27 x+24$
- $y^2=2 x^6+4 x^5+3 x^4+26 x^3+20 x^2+20 x+14$
- $y^2=2 x^6+7 x^5+4 x^4+x^3+8 x^2+11 x+22$
- $y^2=23 x^6+28 x^5+3 x^4+x^3+6 x^2+3 x+2$
- $y^2=11 x^6+24 x^5+14 x^4+18 x^3+16 x^2+25 x+27$
- $y^2=14 x^6+25 x^5+19 x^4+21 x^3+13 x^2+20 x+8$
- $y^2=12 x^6+8 x^5+27 x^4+16 x^3+x^2+4 x+27$
- $y^2=18 x^6+18 x^5+x^4+18 x^3+3 x^2+11 x+4$
- $y^2=26 x^6+2 x^5+14 x^4+21 x^3+x^2+10 x+21$
- $y^2=19 x^6+20 x^5+26 x^4+x^3+19 x^2+21 x+17$
- $y^2=14 x^6+24 x^5+28 x^4+13 x^3+25 x^2+21 x+10$
- $y^2=8 x^6+16 x^5+4 x^4+21 x^2+7 x+21$
- $y^2=3 x^6+8 x^5+27 x^4+18 x^3+12 x^2+12 x+24$
- $y^2=12 x^6+27 x^5+3 x^4+3 x^3+17 x^2+5 x+2$
- $y^2=27 x^6+18 x^5+7 x^4+24 x^3+14 x^2+22 x+14$
- $y^2=x^6+9 x^5+17 x^4+23 x^3+12 x^2+25 x+20$
- $y^2=17 x^6+22 x^5+23 x^4+25 x^3+25 x^2+19$
- $y^2=24 x^6+7 x^5+14 x^4+5 x^3+20 x^2+22 x+7$
- $y^2=18 x^6+5 x^5+6 x^4+4 x^3+11 x^2+17 x+18$
- $y^2=16 x^6+11 x^5+2 x^4+9 x^3+18 x^2+12 x+16$
- $y^2=4 x^6+5 x^5+16 x^4+28 x^3+17 x^2+23 x+8$
- $y^2=16 x^6+7 x^5+9 x^4+13 x^3+x^2+28 x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{29}$.
Endomorphism algebra over $\F_{29}$| The endomorphism algebra of this simple isogeny class is 4.0.4983104.1. |
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 |
|---|---|---|
| 2.29.k_cy | $2$ | (not in LMFDB) |