Invariants
| Base field: | $\F_{47}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 82 x^{2} + 2209 x^{4}$ |
| Frobenius angles: | $\pm0.0813006620146$, $\pm0.918699337985$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{11})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $26$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is simple but not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2128$ | $4528384$ | $10779207376$ | $23788796645376$ | $52599132617621968$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $48$ | $2046$ | $103824$ | $4875070$ | $229345008$ | $10779199422$ | $506623120464$ | $23811295545214$ | $1119130473102768$ | $52599132999413886$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 26 curves (of which all are hyperelliptic):
- $y^2=23 x^6+x^5+6 x^4+7 x^3+23 x^2+16 x+32$
- $y^2=33 x^6+23 x^5+14 x^4+13 x^3+20 x^2+29 x+6$
- $y^2=24 x^6+21 x^5+23 x^4+18 x^3+6 x^2+4 x+30$
- $y^2=24 x^5+30 x^4+x^3+21 x^2+8 x$
- $y^2=39 x^6+12 x^5+18 x^4+31 x^3+11 x^2+x+17$
- $y^2=x^6+29 x^5+7 x^4+20 x^3+21 x^2+6 x+22$
- $y^2=10 x^6+43 x^5+31 x^4+26 x^3+32 x^2+31 x+14$
- $y^2=42 x^6+28 x^5+17 x^4+35 x^3+15 x^2+7 x+30$
- $y^2=19 x^6+45 x^5+8 x^4+34 x^3+35 x^2+25 x+44$
- $y^2=26 x^6+32 x^5+18 x^4+3 x^3+5 x^2+28 x+37$
- $y^2=31 x^6+27 x^5+25 x^4+x^3+7 x^2+14 x+16$
- $y^2=12 x^6+39 x^5+21 x^4+21 x^3+45 x^2+40 x+46$
- $y^2=32 x^6+28 x^5+41 x^4+7 x^3+32 x^2+34 x+30$
- $y^2=29 x^6+45 x^5+33 x^4+25 x^3+9 x^2+19 x+42$
- $y^2=27 x^6+21 x^5+23 x^4+36 x^3+22 x^2+28 x+7$
- $y^2=19 x^6+16 x^5+18 x^4+15 x^3+13 x^2+42 x+37$
- $y^2=37 x^5+6 x^4+34 x^3+29 x+34$
- $y^2=27 x^6+36 x^5+18 x^3+2 x+10$
- $y^2=2 x^6+26 x^5+2 x^4+25 x^3+17 x^2+30$
- $y^2=3 x^6+29 x^5+23 x^4+7 x^3+24 x^2+29 x+44$
- $y^2=14 x^6+24 x^5+11 x^4+37 x^3+43 x^2+38 x+42$
- $y^2=8 x^6+17 x^5+12 x^4+27 x^3+13 x^2+2 x+13$
- $y^2=13 x^6+19 x^5+28 x^4+18 x^3+2 x^2+43 x+42$
- $y^2=9 x^6+36 x^5+3 x^4+46 x^3+35 x^2+13 x+33$
- $y^2=36 x^6+3 x^5+37 x^4+22 x^3+x^2+24 x+18$
- $y^2=16 x^6+10 x^5+25 x^4+31 x^3+23 x^2+x+36$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{47^{2}}$.
Endomorphism algebra over $\F_{47}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{11})\). |
| The base change of $A$ to $\F_{47^{2}}$ is 1.2209.ade 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-33}) \)$)$ |
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 |
|---|---|---|
| 2.47.ag_ch | $3$ | (not in LMFDB) |
| 2.47.g_ch | $3$ | (not in LMFDB) |
| 2.47.a_de | $4$ | (not in LMFDB) |