Normalized defining polynomial
\( x^{4} - x^{3} + 26x^{2} - 26x + 151 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $[0, 2]$ |
| |
| Discriminant: |
\(55125\)
\(\medspace = 3^{2}\cdot 5^{3}\cdot 7^{2}\)
|
| |
| Root discriminant: | \(15.32\) |
| |
| Galois root discriminant: | $3^{1/2}5^{3/4}7^{1/2}\approx 15.322765339111537$ | ||
| Ramified primes: |
\(3\), \(5\), \(7\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_4$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(105=3\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{105}(64,·)$, $\chi_{105}(1,·)$, $\chi_{105}(83,·)$, $\chi_{105}(62,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | 4.0.55125.1$^{2}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{41}a^{3}-\frac{6}{41}a^{2}+\frac{15}{41}a-\frac{19}{41}$
| Monogenic: | No | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}$, which has order $4$ |
| |
| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ |
| |
| Relative class number: | $4$ |
Unit group
| Rank: | $1$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental unit: |
$\frac{1}{41}a^{3}-\frac{6}{41}a^{2}+\frac{15}{41}a-\frac{60}{41}$
|
| |
| Regulator: | \( 0.962423650119 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{2}\cdot 0.962423650119 \cdot 4}{2\cdot\sqrt{55125}}\cr\approx \mathstrut & 0.3236545507052 \end{aligned}\]
Galois group
| A cyclic group of order 4 |
| The 4 conjugacy class representatives for $C_4$ |
| Character table for $C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \) |
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 | ${\href{/padicField/2.4.0.1}{4} }$ | R | R | R | ${\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.2.2a1.1 | $x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
|
\(5\)
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
|
\(7\)
| 7.2.2.2a1.1 | $x^{4} + 12 x^{3} + 42 x^{2} + 43 x + 9$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
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 | 1.105.4t1.a.a | $1$ | $ 3 \cdot 5 \cdot 7 $ | 4.0.55125.1 | $C_4$ (as 4T1) | $0$ | $-1$ |
| *4 | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *4 | 1.105.4t1.a.b | $1$ | $ 3 \cdot 5 \cdot 7 $ | 4.0.55125.1 | $C_4$ (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 *.