Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 4 x + 35 x^{2} - 116 x^{3} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.267088159250$, $\pm0.595904571582$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.978192.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $24$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $757$ | $754729$ | $595005028$ | $500983827097$ | $420984543492997$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $26$ | $896$ | $24398$ | $708324$ | $20524666$ | $594796430$ | $17249409826$ | $500246179780$ | $14507146883654$ | $420707213596736$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=21 x^6+18 x^5+20 x^4+13 x^3+25 x^2+x+10$
- $y^2=x^6+5 x^5+11 x^4+2 x^3+13 x^2+3 x+26$
- $y^2=22 x^6+22 x^5+16 x^4+14 x^3+23 x^2+13 x+21$
- $y^2=12 x^6+18 x^5+2 x^4+18 x^3+23 x^2+10 x+27$
- $y^2=28 x^6+13 x^5+8 x^4+22 x^3+6 x^2+2 x+25$
- $y^2=11 x^6+6 x^5+20 x^3+9 x^2+14 x+3$
- $y^2=12 x^6+16 x^5+18 x^4+19 x^3+x^2+22 x+7$
- $y^2=7 x^6+24 x^5+5 x^4+26 x^3+14 x^2+12 x+26$
- $y^2=4 x^6+19 x^5+2 x^4+2 x^3+25 x+16$
- $y^2=28 x^6+18 x^5+22 x^4+24 x^3+7$
- $y^2=20 x^6+12 x^5+3 x^4+5 x^3+19 x^2+7 x+18$
- $y^2=20 x^6+24 x^5+10 x^4+26 x^3+22 x^2+19 x+25$
- $y^2=18 x^6+23 x^5+5 x^4+19 x^3+9 x^2+20 x+6$
- $y^2=10 x^6+14 x^5+8 x^4+x^3+23 x^2+23$
- $y^2=13 x^6+18 x^5+23 x^4+23 x^3+14 x^2+26 x+3$
- $y^2=9 x^6+4 x^5+11 x^4+14 x^3+3 x^2+24 x+28$
- $y^2=9 x^6+9 x^5+16 x^4+16 x^3+16 x^2+6 x+28$
- $y^2=14 x^6+x^4+21 x^3+20 x^2+4 x+12$
- $y^2=14 x^6+24 x^4+25 x^3+18 x^2+x+23$
- $y^2=8 x^6+11 x^5+8 x^4+15 x^3+26 x^2+4 x+16$
- $y^2=10 x^6+26 x^5+22 x^4+20 x^3+24 x^2+26 x+28$
- $y^2=15 x^6+x^5+9 x^4+26 x^3+2 x^2+22 x+28$
- $y^2=11 x^6+12 x^5+2 x^4+10 x^3+18 x^2+11 x+14$
- $y^2=11 x^6+13 x^5+24 x^4+6 x^3+23 x^2+7 x+5$
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 4.0.978192.1. |
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.e_bj | $2$ | (not in LMFDB) |