Invariants
Base field: | $\F_{5^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 15 x + 103 x^{2} - 375 x^{3} + 625 x^{4}$ |
Frobenius angles: | $\pm0.119565980564$, $\pm0.307050707467$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.153621.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $339$ | $379341$ | $246255363$ | $152923357989$ | $95379168616944$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $11$ | $607$ | $15761$ | $391483$ | $9766826$ | $244132855$ | $6103512461$ | $152588557939$ | $3814703937191$ | $95367466650862$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(4a+3)x^6+(4a+2)x^5+(4a+1)x^4+2ax^3+3x^2+3x+4a$
- $y^2=3ax^6+(4a+2)x^5+ax^4+(4a+3)x^3+(a+2)x^2+(a+3)x$
- $y^2=2ax^6+(2a+2)x^5+(2a+3)x^4+(a+3)x^3+2x^2+4x+4a+3$
- $y^2=(3a+2)x^6+(2a+4)x^5+3ax^4+(4a+4)x^3+(2a+3)x^2+(3a+4)x+a$
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 4.0.153621.1. |
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.p_dz | $2$ | 2.625.at_xl |