Invariants
Base field: | $\F_{5^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 9 x + 25 x^{2} )( 1 - 4 x + 25 x^{2} )$ |
$1 - 13 x + 86 x^{2} - 325 x^{3} + 625 x^{4}$ | |
Frobenius angles: | $\pm0.143566293129$, $\pm0.369010119566$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $8$ |
Isomorphism classes: | 52 |
This isogeny class is not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $374$ | $392700$ | $247003064$ | $152738308800$ | $95352068016854$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $13$ | $629$ | $15808$ | $391009$ | $9764053$ | $244144514$ | $6103712173$ | $152589360769$ | $3814701721168$ | $95367424578149$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+4)x^6+(2a+4)x^5+(3a+2)x^4+(a+2)x^3+(a+3)x^2+2x+3a+3$
- $y^2=3ax^6+(3a+2)x^5+ax^4+(a+3)x^3+(2a+1)x^2+(4a+3)x+4a+1$
- $y^2=(4a+2)x^6+4x^5+(2a+3)x^4+(3a+4)x^3+3x^2+a+2$
- $y^2=(2a+3)x^6+2x^5+(2a+1)x^4+(3a+4)x^3+2x^2+(a+3)x+3a+2$
- $y^2=(3a+2)x^6+3x^5+(2a+1)x^4+(4a+2)x^3+(4a+1)x^2+(a+2)x+3a+2$
- $y^2=(2a+2)x^6+3x^5+(a+4)x^4+(4a+4)x^2+(a+4)x+4a+2$
- $y^2=(3a+2)x^6+ax^5+(a+4)x^4+(4a+2)x^3+(3a+1)x^2+(4a+2)x+a$
- $y^2=(3a+1)x^6+(2a+2)x^5+(2a+3)x^4+x^3+(a+3)x^2+ax$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{2}}$.
Endomorphism algebra over $\F_{5^{2}}$The isogeny class factors as 1.25.aj $\times$ 1.25.ae and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
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.af_o | $2$ | 2.625.d_ho |
2.25.f_o | $2$ | 2.625.d_ho |
2.25.n_di | $2$ | 2.625.d_ho |