Invariants
Base field: | $\F_{5}$ |
Dimension: | $3$ |
L-polynomial: | $1 + x + 6 x^{2} + 9 x^{3} + 30 x^{4} + 25 x^{5} + 125 x^{6}$ |
Frobenius angles: | $\pm0.302174739143$, $\pm0.507824215851$, $\pm0.785429339113$ |
Angle rank: | $3$ (numerical) |
Number field: | 6.0.4943659.1 |
Galois group: | $A_4\times C_2$ |
Jacobians: | $9$ |
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: | $3$ |
Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $197$ | $25019$ | $2130752$ | $257420491$ | $29575945097$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $7$ | $37$ | $136$ | $661$ | $3027$ | $15820$ | $77511$ | $388293$ | $1958248$ | $9772317$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which 0 are hyperelliptic):
- $3 x^4+x^3 y+4 x^2 y^2+x^2 y z+2 x y^2 z+x z^3+y^4=0$
- $2 x^4+3 x^3 y+4 x^2 y^2+2 x^2 y z+2 x y^2 z+x z^3+y^4=0$
- $x^4+3 x^3 z+4 x^2 y^2+x^2 y z+x^2 z^2+x z^3+y^3 z=0$
- $4 x^3 z+3 x^2 y^2+2 x^2 y z+x^2 z^2+x z^3+y^3 z=0$
- $2 x^4+x^3 y+x^3 z+4 x^2 y^2+3 x^2 y z+x^2 z^2+x z^3+y^3 z=0$
- $4 x^3 y+x^2 y^2+2 x^2 y z+x^2 z^2+x y z^2+x z^3+y^3 z=0$
- $3 x^4+3 x^3 z+x^2 y^2+x^2 y z+4 x^2 z^2+x y z^2+x z^3+y^3 z=0$
- $x^4+x^3 y+x^3 z+4 x^2 y^2+x^2 y z+4 x^2 z^2+x y z^2+x z^3+y^3 z=0$
- $2 x^4+4 x^3 z+2 x^2 y^2+x^2 z^2+x y^3+x z^3+y^2 z^2=0$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5}$.
Endomorphism algebra over $\F_{5}$The endomorphism algebra of this simple isogeny class is 6.0.4943659.1. |
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 |
---|---|---|
3.5.ab_g_aj | $2$ | 3.25.l_da_sl |