Invariants
| Base field: | $\F_{47}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 12 x + 47 x^{2} )^{2}$ |
| $1 + 24 x + 238 x^{2} + 1128 x^{3} + 2209 x^{4}$ | |
| Frobenius angles: | $\pm0.839263688900$, $\pm0.839263688900$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $10$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 3, 5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3600$ | $4665600$ | $10786899600$ | $23830018560000$ | $52587799992090000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $72$ | $2110$ | $103896$ | $4883518$ | $229295592$ | $10779628030$ | $506620490616$ | $23811298823038$ | $1119130450770312$ | $52599131932239550$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=25 x^6+24 x^4+24 x^2+25$
- $y^2=7 x^6+20 x^4+20 x^2+7$
- $y^2=37 x^6+40 x^5+11 x^4+13 x^3+40 x^2+43 x+25$
- $y^2=22 x^6+42 x^5+33 x^4+3 x^3+33 x^2+42 x+22$
- $y^2=25 x^6+6 x^5+16 x^4+22 x^3+42 x^2+34 x+37$
- $y^2=32 x^6+19 x^5+44 x^4+18 x^3+44 x^2+19 x+32$
- $y^2=12 x^6+6 x^5+28 x^4+33 x^3+12 x^2+27 x+3$
- $y^2=2 x^6+36 x^5+46 x^4+3 x^3+46 x^2+36 x+2$
- $y^2=29 x^6+4 x^4+4 x^2+29$
- $y^2=16 x^6+x^5+26 x^4+20 x^3+5 x^2+4 x+34$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{47}$.
Endomorphism algebra over $\F_{47}$| The isogeny class factors as 1.47.m 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$ |
Base change
This is a primitive isogeny class.