Invariants
Base field: | $\F_{5^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 13 x + 89 x^{2} - 325 x^{3} + 625 x^{4}$ |
Frobenius angles: | $\pm0.188181882908$, $\pm0.344353995667$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.45461.1 |
Galois group: | $D_{4}$ |
Jacobians: | $9$ |
Isomorphism classes: | 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: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $377$ | $396981$ | $248856569$ | $153120732453$ | $95388246850832$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $13$ | $635$ | $15925$ | $391987$ | $9767758$ | $244138763$ | $6103557109$ | $152588437987$ | $3814698751357$ | $95367416886350$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 9 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(a+2)x^6+(2a+1)x^5+2x^4+2ax^3+2ax^2+3ax+a+2$
- $y^2=2ax^6+(3a+4)x^5+(2a+4)x^4+4ax^3+2ax^2+x+3a+1$
- $y^2=(a+2)x^6+x^5+(3a+4)x^4+(2a+3)x^3+(4a+3)x^2+(4a+4)x+2a+2$
- $y^2=3ax^6+2ax^5+2ax^4+(2a+3)x^3+(2a+3)x^2+2x+4a+1$
- $y^2=(2a+1)x^6+(4a+1)x^5+(a+1)x^4+(4a+1)x^3+(3a+3)x^2+(2a+1)x+a+1$
- $y^2=(2a+4)x^6+2x^5+2ax^4+(3a+1)x^3+(4a+1)x^2+(3a+3)x+2a+4$
- $y^2=(4a+4)x^6+2ax^5+4ax^4+(3a+1)x^3+(a+4)x^2+(a+1)x+3a+1$
- $y^2=3ax^6+2ax^5+2ax^4+(4a+2)x^3+ax^2+2x+a+1$
- $y^2=(2a+3)x^6+(a+4)x^4+4ax^3+3ax^2+(2a+3)x+a$
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 4.0.45461.1. |
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_dl | $2$ | 2.625.j_bbt |