Invariants
Base field: | $\F_{5^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 12 x + 79 x^{2} - 300 x^{3} + 625 x^{4}$ |
Frobenius angles: | $\pm0.167587632578$, $\pm0.391120132964$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.195216.2 |
Galois group: | $D_{4}$ |
Jacobians: | $10$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $393$ | $399681$ | $247547556$ | $152739293193$ | $95360268956793$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $14$ | $640$ | $15842$ | $391012$ | $9764894$ | $244157398$ | $6103755518$ | $152588983492$ | $3814696917650$ | $95367402613600$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+(a+3)x^5+(3a+1)x^4+(2a+2)x^3+(a+1)x+3a+3$
- $y^2=(3a+1)x^6+(a+4)x^5+3x^4+(2a+2)x^3+(2a+4)x+4a+1$
- $y^2=(4a+3)x^6+(4a+2)x^5+(4a+2)x^4+(3a+2)x^3+(4a+3)x^2+(a+1)x+2a+1$
- $y^2=2ax^6+(a+2)x^5+4ax^4+(4a+4)x^3+(a+4)x^2+4ax+4a+2$
- $y^2=4ax^6+(4a+4)x^5+(4a+4)x^4+(3a+3)x^3+(4a+1)x+2a$
- $y^2=2x^6+(4a+3)x^5+ax^4+(4a+3)x^3+(4a+2)x+2$
- $y^2=(2a+1)x^6+(a+4)x^5+2x^4+(2a+2)x^3+4x^2+(2a+2)x+4a$
- $y^2=2ax^6+2ax^5+(2a+2)x^4+(3a+1)x^3+2ax^2+(a+3)x+a$
- $y^2=3ax^6+(2a+3)x^5+(3a+2)x^4+(a+2)x^3+(3a+1)x^2+(3a+2)x+3a$
- $y^2=(4a+3)x^6+(3a+1)x^5+ax^4+2x^3+(a+1)x^2+(2a+3)x+3a+1$
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 4.0.195216.2. |
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.m_db | $2$ | 2.625.o_lf |