Invariants
Base field: | $\F_{5^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 13 x + 85 x^{2} - 325 x^{3} + 625 x^{4}$ |
Frobenius angles: | $\pm0.128789547124$, $\pm0.375668685983$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.44573.1 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $373$ | $391277$ | $246386269$ | $152607811925$ | $95337474393808$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $13$ | $627$ | $15769$ | $390675$ | $9762558$ | $244142427$ | $6103726369$ | $152589449475$ | $3814702120333$ | $95367429615502$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+(2a+2)x^5+(a+2)x^4+(3a+2)x^3+(4a+1)x^2+(3a+4)x+4$
- $y^2=(4a+2)x^6+4ax^5+(3a+2)x^4+(a+2)x^3+(3a+1)x^2+(a+3)x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{2}}$.
Endomorphism algebra over $\F_{5^{2}}$The endomorphism algebra of this simple isogeny class is 4.0.44573.1. |
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 |
---|---|---|
2.25.n_dh | $2$ | 2.625.b_z |