Normalized defining polynomial
\( x^{5} + 10x^{3} - 25x^{2} + 15x + 6 \)
Invariants
| Degree: | $5$ |
| |
| Signature: | $(1, 2)$ |
| |
| Discriminant: |
\(16078125\)
\(\medspace = 3\cdot 5^{6}\cdot 7^{3}\)
|
| |
| Root discriminant: | \(27.62\) |
| |
| Galois root discriminant: | $3^{1/2}5^{13/10}7^{5/6}\approx 71.034609880671$ | ||
| Ramified primes: |
\(3\), \(5\), \(7\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{21}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{17}a^{4}+\frac{8}{17}a^{3}+\frac{6}{17}a^{2}+\frac{6}{17}a-\frac{5}{17}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
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: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{37}{17}a^{4}-\frac{61}{17}a^{3}+\frac{154}{17}a^{2}-\frac{1937}{17}a-\frac{661}{17}$, $\frac{771}{17}a^{4}+\frac{2258}{17}a^{3}+\frac{11715}{17}a^{2}+\frac{8910}{17}a+\frac{1585}{17}$
|
| |
| Regulator: | \( 184.634692508 \) |
| |
| Unit signature rank: | \( 1 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{2}\cdot 184.634692508 \cdot 1}{2\cdot\sqrt{16078125}}\cr\approx \mathstrut & 1.81783869541 \end{aligned}\]
Galois group
| A non-solvable group of order 120 |
| The 7 conjugacy class representatives for $S_5$ |
| Character table for $S_5$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 6 sibling: | 6.2.7090453125.1 |
| Degree 10 siblings: | 10.2.5428628173828125.1, deg 10 |
| Degree 12 sibling: | deg 12 |
| Degree 15 sibling: | deg 15 |
| Degree 20 siblings: | deg 20, deg 20, deg 20 |
| Degree 24 sibling: | deg 24 |
| Degree 30 siblings: | deg 30, deg 30, deg 30 |
| Degree 40 sibling: | deg 40 |
| 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 | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | R | R | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
|
\(5\)
| 5.1.5.6a1.1 | $x^{5} + 10 x^{2} + 5$ | $5$ | $1$ | $6$ | $D_{5}$ | $$[\frac{3}{2}]_{2}$$ |
|
\(7\)
| 7.1.2.1a1.1 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 7.1.3.2a1.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *120 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.21.2t1.a.a | $1$ | $ 3 \cdot 7 $ | \(\Q(\sqrt{21}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 4.144703125.10t12.a.a | $4$ | $ 3^{3} \cdot 5^{6} \cdot 7^{3}$ | 5.1.16078125.1 | $S_5$ (as 5T5) | $1$ | $0$ | |
| *120 | 4.16078125.5t5.a.a | $4$ | $ 3 \cdot 5^{6} \cdot 7^{3}$ | 5.1.16078125.1 | $S_5$ (as 5T5) | $1$ | $0$ |
| 5.337640625.10t13.a.a | $5$ | $ 3^{2} \cdot 5^{6} \cdot 7^{4}$ | 5.1.16078125.1 | $S_5$ (as 5T5) | $1$ | $1$ | |
| 5.7090453125.6t14.a.a | $5$ | $ 3^{3} \cdot 5^{6} \cdot 7^{5}$ | 5.1.16078125.1 | $S_5$ (as 5T5) | $1$ | $1$ | |
| 6.177261328125.20t30.b.a | $6$ | $ 3^{3} \cdot 5^{8} \cdot 7^{5}$ | 5.1.16078125.1 | $S_5$ (as 5T5) | $1$ | $-2$ |
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 *.