Normalized defining polynomial
\( x^{4} - x^{3} - 23x^{2} + x + 86 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $[4, 0]$ |
| |
| Discriminant: |
\(122825\)
\(\medspace = 5^{2}\cdot 17^{3}\)
|
| |
| Root discriminant: | \(18.72\) |
| |
| Galois root discriminant: | $5^{1/2}17^{3/4}\approx 18.720683165352195$ | ||
| Ramified primes: |
\(5\), \(17\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_4$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(85=5\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{85}(64,·)$, $\chi_{85}(1,·)$, $\chi_{85}(4,·)$, $\chi_{85}(16,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{3}-\frac{1}{4}a^{2}+\frac{3}{8}a-\frac{1}{4}$
| Monogenic: | No | |
| Index: | $4$ | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{13}{4}a-\frac{3}{2}$, $\frac{1}{2}a^{3}-2a^{2}-\frac{11}{2}a+16$, $\frac{1}{2}a^{3}-a^{2}-\frac{17}{2}a+6$
|
| |
| Regulator: | \( 61.2253005252 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{0}\cdot 61.2253005252 \cdot 2}{2\cdot\sqrt{122825}}\cr\approx \mathstrut & 2.79516546464 \end{aligned}\]
Galois group
| A cyclic group of order 4 |
| The 4 conjugacy class representatives for $C_4$ |
| Character table for $C_4$ |
Intermediate fields
| \(\Q(\sqrt{17}) \) |
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 | ${\href{/padicField/2.1.0.1}{1} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }$ | R | ${\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.2.2a1.1 | $x^{4} + 8 x^{3} + 20 x^{2} + 21 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
|
\(17\)
| 17.1.4.3a1.3 | $x^{4} + 153$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
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 | 1.85.4t1.a.a | $1$ | $ 5 \cdot 17 $ | 4.4.122825.1 | $C_4$ (as 4T1) | $0$ | $1$ |
| *4 | 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *4 | 1.85.4t1.a.b | $1$ | $ 5 \cdot 17 $ | 4.4.122825.1 | $C_4$ (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 *.