Normalized defining polynomial
\( x^{10} - x^{9} + 6x^{8} - 3x^{7} + 11x^{6} - 3x^{5} + 11x^{4} - 3x^{3} + 6x^{2} - x + 1 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-229345007\) \(\medspace = -\,47^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(6.86\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $47^{1/2}\approx 6.855654600401044$ | ||
Ramified primes: | \(47\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-47}) \) | ||
$\card{ \Gal(K/\Q) }$: | $10$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5}a^{9}+\frac{1}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{1}{5}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{1}{5}a^{9}+a^{8}+\frac{1}{5}a^{7}+\frac{28}{5}a^{6}-\frac{1}{5}a^{5}+\frac{41}{5}a^{4}-\frac{3}{5}a^{3}+\frac{34}{5}a^{2}-2a+\frac{14}{5}$, $\frac{4}{5}a^{9}-2a^{8}+\frac{24}{5}a^{7}-\frac{43}{5}a^{6}+\frac{31}{5}a^{5}-\frac{66}{5}a^{4}+\frac{18}{5}a^{3}-\frac{64}{5}a^{2}+2a-\frac{19}{5}$, $\frac{3}{5}a^{9}+\frac{18}{5}a^{7}+\frac{4}{5}a^{6}+\frac{37}{5}a^{5}+\frac{8}{5}a^{4}+\frac{31}{5}a^{3}+\frac{2}{5}a^{2}+2a-\frac{3}{5}$, $\frac{8}{5}a^{9}-a^{8}+\frac{43}{5}a^{7}-\frac{6}{5}a^{6}+\frac{67}{5}a^{5}+\frac{3}{5}a^{4}+\frac{56}{5}a^{3}-\frac{8}{5}a^{2}+4a-\frac{3}{5}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 0.60186322759 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 0.60186322759 \cdot 1}{2\cdot\sqrt{229345007}}\cr\approx \mathstrut & 0.194590892984 \end{aligned}\]
Galois group
A solvable group of order 10 |
The 4 conjugacy class representatives for $D_5$ |
Character table for $D_5$ |
Intermediate fields
\(\Q(\sqrt{-47}) \), 5.1.2209.1 x5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.2209.1 |
Minimal sibling: | 5.1.2209.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{5}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{5}$ | ${\href{/padicField/13.2.0.1}{2} }^{5}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{5}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}$ | ${\href{/padicField/29.2.0.1}{2} }^{5}$ | ${\href{/padicField/31.2.0.1}{2} }^{5}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{5}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}$ | R | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(47\) | 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |