Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + x + 29 x^{2} )( 1 + 4 x + 29 x^{2} )$ |
| $1 + 5 x + 62 x^{2} + 145 x^{3} + 841 x^{4}$ | |
| Frobenius angles: | $\pm0.529596959677$, $\pm0.621118941591$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $10$ |
| Isomorphism classes: | 26 |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1054$ | $794716$ | $585872224$ | $499081648000$ | $420966262369654$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $35$ | $941$ | $24020$ | $705633$ | $20523775$ | $594832826$ | $17249596315$ | $500246779873$ | $14507149505300$ | $420707225510981$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=20 x^6+6 x^5+26 x^4+12 x^3+22 x^2+24 x+20$
- $y^2=4 x^6+8 x^3+9 x^2+11 x+11$
- $y^2=16 x^6+4 x^5+6 x^4+26 x^3+25 x^2+24 x+11$
- $y^2=19 x^6+10 x^5+13 x^4+3 x^3+26 x^2+6$
- $y^2=20 x^6+12 x^5+17 x^3+16 x^2+22 x+5$
- $y^2=10 x^6+28 x^5+25 x^4+7 x^3+9 x^2+4 x+17$
- $y^2=14 x^6+25 x^5+x^4+14 x^3+25 x^2+6 x+12$
- $y^2=13 x^6+27 x^5+28 x^4+28 x^3+21 x^2+28 x+26$
- $y^2=28 x^6+20 x^5+23 x^4+11 x^3+16 x^2+9 x+14$
- $y^2=20 x^6+10 x^5+26 x^4+26 x^3+22 x^2+x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{29}$.
Endomorphism algebra over $\F_{29}$| The isogeny class factors as 1.29.b $\times$ 1.29.e and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.