Invariants
Base field: | $\F_{5^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 14 x + 94 x^{2} - 350 x^{3} + 625 x^{4}$ |
Frobenius angles: | $\pm0.125226661552$, $\pm0.341943888296$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.28400.1 |
Galois group: | $D_{4}$ |
Jacobians: | $12$ |
Isomorphism classes: | 16 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $356$ | $385904$ | $246554564$ | $152787111680$ | $95353633346436$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $12$ | $618$ | $15780$ | $391134$ | $9764212$ | $244131978$ | $6103609020$ | $152589198654$ | $3814704655932$ | $95367450243178$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(a+4)x^6+3ax^5+(a+2)x^4+(2a+2)x^3+(3a+2)x^2+3a+1$
- $y^2=4ax^6+(2a+4)x^5+(3a+4)x^4+(4a+4)x^3+(2a+1)x^2+(a+1)x+2a+3$
- $y^2=2ax^6+3ax^5+(4a+2)x^4+(2a+1)x^3+(4a+3)x^2+4ax+4a+1$
- $y^2=(3a+3)x^6+4x^5+2ax^4+ax^3+(4a+3)x^2+(4a+1)x+2a+3$
- $y^2=3ax^6+(4a+3)x^5+(4a+3)x^4+(a+1)x^3+4ax^2+(3a+3)x+3a$
- $y^2=2ax^6+(a+2)x^5+(4a+2)x^4+(3a+4)x^3+(2a+3)x^2+(4a+2)x+3a+4$
- $y^2=ax^6+(2a+2)x^5+3x^4+(a+2)x^3+4ax^2+(4a+1)x+3a+1$
- $y^2=ax^6+(3a+2)x^5+(4a+4)x^4+(2a+2)x^3+(4a+1)x^2+(4a+1)x+4a+1$
- $y^2=(4a+1)x^6+(3a+3)x^5+4x^4+x^3+(a+2)x^2+3ax+a$
- $y^2=(a+4)x^6+ax^5+4ax^4+2x^3+(4a+1)x^2+2ax+4a$
- $y^2=2ax^6+(4a+4)x^5+4x^4+3x^3+(4a+4)x^2+(3a+4)x+4a$
- $y^2=(4a+1)x^6+(a+2)x^5+4ax^4+(a+3)x^3+x^2+(3a+4)x+3a+2$
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 4.0.28400.1. |
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.o_dq | $2$ | 2.625.ai_la |