Invariants
Base field: | $\F_{5}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 3 x + 5 x^{2} )( 1 + x + 5 x^{2} )$ |
$1 - 2 x + 7 x^{2} - 10 x^{3} + 25 x^{4}$ | |
Frobenius angles: | $\pm0.265942140215$, $\pm0.571783146564$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $6$ |
Isomorphism classes: | 10 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $21$ | $945$ | $16128$ | $401625$ | $10271541$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $36$ | $130$ | $644$ | $3284$ | $15606$ | $77060$ | $389764$ | $1954714$ | $9764676$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+x^5+3x^4+3x^3+3x^2+x+1$
- $y^2=2x^6+2x^5+3x^4+3x+3$
- $y^2=3x^6+4x^5+2x^4+x+2$
- $y^2=2x^6+3x^5+x^3+3x+3$
- $y^2=4x^6+2x^5+2x^4+x^2+3x+1$
- $y^2=4x^6+2x^5+x^4+3x^3+x^2+2x+4$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5}$.
Endomorphism algebra over $\F_{5}$The isogeny class factors as 1.5.ad $\times$ 1.5.b 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 |
---|---|---|
2.5.ae_n | $2$ | 2.25.k_ch |
2.5.c_h | $2$ | 2.25.k_ch |
2.5.e_n | $2$ | 2.25.k_ch |