Invariants
| Base field: | $\F_{5^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 25 x^{2} + 625 x^{4}$ |
| Frobenius angles: | $\pm0.166666666667$, $\pm0.833333333333$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\zeta_{12})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $8$ |
| Cyclic group of points: | yes |
This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $601$ | $361201$ | $244171876$ | $153077345001$ | $95367421875001$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $26$ | $576$ | $15626$ | $391876$ | $9765626$ | $244203126$ | $6103515626$ | $152588671876$ | $3814697265626$ | $95367412109376$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=a x^6+(3 a+1) x^5+a x^4+(a+3) x^2+(2 a+1) x+4 a+3$
- $y^2=(a+3) x^6+(4 a+4) x^5+(a+3) x^4+(4 a+3) x^2+(3 a+1) x+2 a+2$
- $y^2=(3 a+4) x^6+(a+3) x^5+(2 a+1) x^4+3 a x^3+(4 a+2) x^2+(3 a+3) x+a+4$
- $y^2=(2 a+4) x^6+(4 a+3) x^5+(3 a+1) x^4+(3 a+4) x^3+(a+2) x^2+(a+4) x+3$
- $y^2=3 a x^6+(a+4) x^5+x^4+(a+2) x^3+(3 a+3) x^2+4 x+3$
- $y^2=(3 a+4) x^6+3 x^5+a x^4+(3 a+3) x^3+(a+4) x^2+4 a x+3 a$
- $y^2=(2 a+1) x^6+(2 a+2) x^5+(2 a+3) x^4+x^3+(4 a+3) x^2+2 a x+2 a$
- $y^2=(3 a+1) x^6+(4 a+1) x^5+x^4+a x^3+(2 a+2) x^2+(2 a+1) x+2 a+1$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{12}}$.
Endomorphism algebra over $\F_{5^{2}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{12})\). |
| The base change of $A$ to $\F_{5^{12}}$ is 1.244140625.bufy 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$. |
- Endomorphism algebra over $\F_{5^{4}}$
The base change of $A$ to $\F_{5^{4}}$ is 1.625.az 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ - Endomorphism algebra over $\F_{5^{6}}$
The base change of $A$ to $\F_{5^{6}}$ is the simple isogeny class 2.15625.a_bufy and its endomorphism algebra is the quaternion algebra over \(\Q(\sqrt{-1}) \) with the following ramification data at primes above $5$, and unramified at all archimedean places:
where $\pi$ is a root of $x^{2} + 1$.$v$ ($ 5 $,\( \pi + 2 \)) ($ 5 $,\( \pi + 3 \)) $\operatorname{inv}_v$ $1/2$ $1/2$
Base change
This is a primitive isogeny class.