Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 15 x + 110 x^{2} - 435 x^{3} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.152255069213$, $\pm0.331511979000$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.613836.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $502$ | $703804$ | $601444192$ | $501266100096$ | $420754934229502$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $15$ | $837$ | $24660$ | $708721$ | $20513475$ | $594821562$ | $17249993055$ | $500248158913$ | $14507156602260$ | $420707252905677$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=28x^6+x^5+16x^4+24x^3+9x^2+17x+4$
- $y^2=23x^6+25x^5+4x^4+22x^3+7x^2+4x+14$
- $y^2=10x^6+16x^5+16x^4+27x^3+13x^2+17x+19$
- $y^2=12x^5+22x^4+20x^3+7x^2+19x+10$
- $y^2=21x^6+28x^5+17x^4+19x^3+2x^2+x+27$
- $y^2=16x^5+2x^4+6x^3+22x^2+5x+24$
- $y^2=17x^6+2x^5+6x^4+27x^3+27x^2+27x+24$
- $y^2=28x^6+28x^5+3x^4+7x^3+20x^2+19x+17$
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 4.0.613836.1. |
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.p_eg | $2$ | (not in LMFDB) |