Invariants
| Base field: | $\F_{47}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 13 x + 47 x^{2} )( 1 - 10 x + 47 x^{2} )$ |
| $1 - 23 x + 224 x^{2} - 1081 x^{3} + 2209 x^{4}$ | |
| Frobenius angles: | $\pm0.102979434792$, $\pm0.239834262915$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $4$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1330$ | $4705540$ | $10784049640$ | $23826784122400$ | $52605088131130150$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $25$ | $2129$ | $103870$ | $4882857$ | $229370975$ | $10779330026$ | $506623272425$ | $23811285521713$ | $1119130479467650$ | $52599132548499689$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=40 x^6+x^5+29 x^4+46 x^3+38 x^2+23 x+41$
- $y^2=46 x^6+40 x^5+42 x^4+40 x^3+28 x^2+29 x+16$
- $y^2=5 x^6+31 x^5+19 x^4+36 x^3+16 x^2+12 x+27$
- $y^2=40 x^6+9 x^5+6 x^4+13 x^3+46 x^2+14 x+7$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{47}$.
Endomorphism algebra over $\F_{47}$| The isogeny class factors as 1.47.an $\times$ 1.47.ak and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
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.ad_abk | $2$ | (not in LMFDB) |
| 2.47.d_abk | $2$ | (not in LMFDB) |
| 2.47.x_iq | $2$ | (not in LMFDB) |