Invariants
Base field: | $\F_{5^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 14 x + 96 x^{2} - 350 x^{3} + 625 x^{4}$ |
Frobenius angles: | $\pm0.162037477264$, $\pm0.323393582701$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.247104.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $358$ | $388788$ | $247888150$ | $153103159248$ | $95396013782278$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $12$ | $622$ | $15864$ | $391942$ | $9768552$ | $244141054$ | $6103552572$ | $152588568382$ | $3814701528444$ | $95367438919102$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+4)x^6+(3a+3)x^5+(2a+1)x^4+(a+4)x^3+ax^2+(3a+3)x+4a+4$
- $y^2=(a+4)x^6+(4a+2)x^5+ax^4+(a+1)x^3+(2a+4)x^2+4ax+2a+4$
- $y^2=(a+2)x^6+(4a+1)x^4+(2a+4)x^3+(3a+3)x^2+(a+1)x+3a$
- $y^2=(4a+1)x^6+3x^5+(4a+4)x^4+ax^2+4x+3a+1$
- $y^2=(4a+1)x^6+(3a+3)x^5+x^4+(2a+4)x^3+4ax^2+3x+a+4$
- $y^2=(3a+2)x^5+(2a+1)x^4+(3a+2)x^3+(3a+2)x^2+(a+3)x$
- $y^2=(3a+1)x^6+(4a+2)x^5+3x^4+(a+3)x^3+3ax^2+(4a+3)x+a+1$
- $y^2=(2a+1)x^6+x^5+(3a+1)x^4+(a+4)x^3+(3a+3)x^2+(a+2)x+3a$
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 4.0.247104.1. |
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_ds | $2$ | 2.625.ae_zq |