Normalized defining polynomial
\( x^{4} - x^{3} - 15x^{2} - 18x - 4 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $[4, 0]$ |
| |
| Discriminant: |
\(68921\)
\(\medspace = 41^{3}\)
|
| |
| Root discriminant: | \(16.20\) |
| |
| Galois root discriminant: | $41^{3/4}\approx 16.202718714300598$ | ||
| Ramified primes: |
\(41\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{41}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_4$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{41}(32,·)$, $\chi_{41}(1,·)$, $\chi_{41}(9,·)$, $\chi_{41}(40,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$
| Monogenic: | No | |
| Index: | $2$ | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a+1$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{15}{2}a-8$, $\frac{3}{2}a^{3}-\frac{3}{2}a^{2}-\frac{47}{2}a-23$
|
| |
| Regulator: | \( 13.5247244386 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{0}\cdot 13.5247244386 \cdot 1}{2\cdot\sqrt{68921}}\cr\approx \mathstrut & 0.412137969577 \end{aligned}\]
Galois group
| A cyclic group of order 4 |
| The 4 conjugacy class representatives for $C_4$ |
| Character table for $C_4$ |
Intermediate fields
| \(\Q(\sqrt{41}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(41\)
| 41.1.4.3a1.1 | $x^{4} + 41$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *4 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *4 | 1.41.4t1.a.a | $1$ | $ 41 $ | 4.4.68921.1 | $C_4$ (as 4T1) | $0$ | $1$ |
| *4 | 1.41.2t1.a.a | $1$ | $ 41 $ | \(\Q(\sqrt{41}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *4 | 1.41.4t1.a.b | $1$ | $ 41 $ | 4.4.68921.1 | $C_4$ (as 4T1) | $0$ | $1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.