Invariants
Base field: | $\F_{5^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 13 x + 81 x^{2} - 325 x^{3} + 625 x^{4}$ |
Frobenius angles: | $\pm0.0544401465072$, $\pm0.398133042327$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.6525.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 10 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $369$ | $385605$ | $243924129$ | $152070658245$ | $95266422092304$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $13$ | $619$ | $15613$ | $389299$ | $9755278$ | $244113739$ | $6103581133$ | $152588386339$ | $3814696133053$ | $95367414858574$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=ax^6+4ax^4+4ax^3+ax^2+ax$
- $y^2=4ax^6+3x^5+3ax^4+(3a+1)x^3+4ax^2+(3a+4)x+a+3$
- $y^2=(2a+4)x^6+(3a+1)x^5+2ax^4+(4a+1)x^3+ax^2+3ax+a+4$
- $y^2=4ax^6+(3a+4)x^5+(a+4)x^4+3x^3+(a+1)x^2+2x+3a+1$
- $y^2=(3a+2)x^6+ax^5+x^4+(a+2)x^3+4ax^2+(a+1)x+2a+4$
- $y^2=(2a+3)x^6+(2a+1)x^5+(3a+1)x^4+ax^3+3ax^2+2ax+a$
- $y^2=(3a+1)x^6+4x^5+(a+4)x^4+(4a+3)x^3+(a+2)x^2+(3a+3)x+2a+4$
- $y^2=2ax^6+2ax^5+2x^4+(2a+4)x^3+4ax^2+(4a+1)x+4a+3$
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 4.0.6525.1. |
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_dd | $2$ | 2.625.ah_ayp |