Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 5 x + 4 x^{2} - 145 x^{3} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.0981767877359$, $\pm0.662482234425$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-198 +10 \sqrt{241}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $18$ |
| Isomorphism classes: | 18 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $696$ | $693216$ | $582735744$ | $500393810304$ | $420805213757976$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $25$ | $825$ | $23890$ | $707489$ | $20515925$ | $594788166$ | $17250131945$ | $500248350049$ | $14507145525370$ | $420707289932625$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):
- $y^2=18 x^6+7 x^5+24 x^4+3 x^3+24 x^2+10 x+22$
- $y^2=14 x^6+25 x^5+5 x^4+15 x^3+5 x^2+20 x+12$
- $y^2=6 x^6+26 x^5+13 x^4+4 x^3+14 x^2+x+8$
- $y^2=14 x^6+19 x^5+21 x^4+16 x^3+26 x^2+14 x+15$
- $y^2=3 x^6+5 x^5+26 x^4+16 x^2+11 x+26$
- $y^2=14 x^6+2 x^5+14 x^4+20 x^3+8 x^2+6 x$
- $y^2=12 x^6+22 x^5+10 x^4+10 x^3+26 x^2+25 x+27$
- $y^2=8 x^6+28 x^5+18 x^4+24 x^3+19 x^2+25 x+13$
- $y^2=23 x^6+22 x^5+21 x^4+27 x^3+27 x^2+2 x+25$
- $y^2=24 x^6+5 x^5+28 x^3+7 x^2+26 x+18$
- $y^2=20 x^6+4 x^5+26 x^4+19 x^3+27 x^2+26 x+5$
- $y^2=17 x^6+3 x^5+25 x^4+20 x^3+27 x^2+10 x+11$
- $y^2=26 x^6+15 x^5+5 x^4+27 x^3+28 x^2+9 x+25$
- $y^2=8 x^6+19 x^5+27 x^4+27 x^2+17 x+3$
- $y^2=23 x^6+5 x^5+13 x^4+21 x^3+25 x^2+26 x+10$
- $y^2=2 x^5+23 x^4+20 x^3+16 x^2+15 x+18$
- $y^2=21 x^6+21 x^5+10 x^4+25 x^3+23 x^2+25 x+25$
- $y^2=6 x^6+7 x^5+6 x^4+17 x^3+23 x^2+x+11$
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 \(\Q(\sqrt{-198 +10 \sqrt{241}})\). |
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.f_e | $2$ | (not in LMFDB) |