Normalized defining polynomial
\( x^{4} - x^{3} + x^{2} + 24x + 4 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $(2, 1)$ |
| |
| Discriminant: |
\(-136552\)
\(\medspace = -\,2^{3}\cdot 13^{2}\cdot 101\)
|
| |
| Root discriminant: | \(19.22\) |
| |
| Galois root discriminant: | $2^{3/2}13^{1/2}101^{1/2}\approx 102.48902380255166$ | ||
| Ramified primes: |
\(2\), \(13\), \(101\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-202}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{8}a^{3}+\frac{1}{8}a^{2}+\frac{3}{8}a-\frac{1}{4}$
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
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: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{3}{8}a^{3}-\frac{5}{8}a^{2}+\frac{9}{8}a+\frac{29}{4}$, $\frac{3}{8}a^{3}-\frac{5}{8}a^{2}+\frac{1}{8}a+\frac{37}{4}$
|
| |
| Regulator: | \( 11.3375316424 \) |
| |
| Unit signature rank: | \( 2 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{1}\cdot 11.3375316424 \cdot 2}{2\cdot\sqrt{136552}}\cr\approx \mathstrut & 0.771097578240 \end{aligned}\]
Galois group
| A solvable group of order 24 |
| The 5 conjugacy class representatives for $S_4$ |
| Character table for $S_4$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Galois closure: | deg 24 |
| Degree 6 siblings: | 6.2.110334016.2, 6.0.89149884928.2 |
| Degree 8 sibling: | 8.0.12173595086688256.11 |
| Degree 12 siblings: | deg 12, deg 12 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.3a1.1 | $x^{2} + 2$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ |
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
|
\(13\)
| 13.2.2.2a1.1 | $x^{4} + 24 x^{3} + 148 x^{2} + 61 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
|
\(101\)
| 101.2.1.0a1.1 | $x^{2} + 97 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 101.1.2.1a1.1 | $x^{2} + 101$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *24 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.808.2t1.b.a | $1$ | $ 2^{3} \cdot 101 $ | \(\Q(\sqrt{-202}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.808.3t2.a.a | $2$ | $ 2^{3} \cdot 101 $ | 3.1.808.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 3.110334016.6t8.a.a | $3$ | $ 2^{6} \cdot 13^{2} \cdot 101^{2}$ | 4.2.136552.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| *24 | 3.136552.4t5.a.a | $3$ | $ 2^{3} \cdot 13^{2} \cdot 101 $ | 4.2.136552.1 | $S_4$ (as 4T5) | $1$ | $1$ |
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 *.