Normalized defining polynomial
\( x^{6} - 15x^{4} - 17x^{3} + 3x^{2} + 21x + 19 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(4, 1)$ |
| |
| Discriminant: |
\(-4435591023\)
\(\medspace = -\,3^{6}\cdot 17\cdot 71^{3}\)
|
| |
| Root discriminant: | \(40.53\) |
| |
| Galois root discriminant: | $3^{43/36}17^{1/2}71^{1/2}\approx 129.0471646971052$ | ||
| Ramified primes: |
\(3\), \(17\), \(71\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-1207}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{46}a^{5}+\frac{9}{46}a^{4}+\frac{10}{23}a^{3}-\frac{21}{46}a^{2}-\frac{1}{23}a+\frac{3}{46}$
| 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: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{2}{23}a^{5}-\frac{5}{23}a^{4}-\frac{29}{23}a^{3}+\frac{27}{23}a^{2}+\frac{42}{23}a+\frac{52}{23}$, $\frac{25}{23}a^{5}-\frac{5}{23}a^{4}-\frac{397}{23}a^{3}-\frac{318}{23}a^{2}+\frac{479}{23}a+\frac{397}{23}$, $\frac{4}{23}a^{5}+\frac{13}{23}a^{4}-\frac{58}{23}a^{3}-\frac{245}{23}a^{2}-\frac{238}{23}a-\frac{149}{23}$, $\frac{11}{46}a^{5}+\frac{7}{46}a^{4}-\frac{74}{23}a^{3}-\frac{277}{46}a^{2}-\frac{149}{23}a-\frac{197}{46}$
|
| |
| Regulator: | \( 865.4820841 \) |
| |
| Unit signature rank: | \( 3 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{1}\cdot 865.4820841 \cdot 1}{2\cdot\sqrt{4435591023}}\cr\approx \mathstrut & 0.6532090447 \end{aligned}\]
Galois group
$\SOPlus(4,2)$ (as 6T13):
| A solvable group of order 72 |
| The 9 conjugacy class representatives for $C_3^2:D_4$ |
| Character table for $C_3^2:D_4$ |
Intermediate fields
| \(\Q(\sqrt{213}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 6.0.254291967.2 |
| Degree 6 sibling: | 6.0.254291967.2 |
| Degree 9 sibling: | deg 9 |
| Degree 12 siblings: | deg 12, deg 12, deg 12, deg 12, deg 12, deg 12 |
| Degree 18 siblings: | deg 18, deg 18, deg 18 |
| Degree 24 siblings: | deg 24, deg 24 |
| Degree 36 siblings: | deg 36, deg 36, deg 36 |
| Minimal sibling: | 6.0.254291967.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }$ | R | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.6.6a1.1 | $x^{6} + 3 x + 3$ | $6$ | $1$ | $6$ | $C_3^2:D_4$ | $$[\frac{5}{4}, \frac{5}{4}]_{4}^{2}$$ |
|
\(17\)
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 17.2.1.0a1.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 17.1.2.1a1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(71\)
| 71.3.2.3a1.1 | $x^{6} + 8 x^{4} + 128 x^{3} + 16 x^{2} + 583 x + 4096$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *72 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.1207.2t1.a.a | $1$ | $ 17 \cdot 71 $ | \(\Q(\sqrt{-1207}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *72 | 1.213.2t1.a.a | $1$ | $ 3 \cdot 71 $ | \(\Q(\sqrt{213}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.51.2t1.a.a | $1$ | $ 3 \cdot 17 $ | \(\Q(\sqrt{-51}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.10863.4t3.a.a | $2$ | $ 3^{2} \cdot 17 \cdot 71 $ | 4.0.554013.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| 4.6018243219.12t34.b.a | $4$ | $ 3^{5} \cdot 17^{3} \cdot 71^{2}$ | 6.4.4435591023.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
| *72 | 4.20824371.6t13.b.a | $4$ | $ 3^{5} \cdot 17 \cdot 71^{2}$ | 6.4.4435591023.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ |
| 4.4986117.6t13.b.a | $4$ | $ 3^{5} \cdot 17^{2} \cdot 71 $ | 6.4.4435591023.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.25135015797.12t34.b.a | $4$ | $ 3^{5} \cdot 17^{2} \cdot 71^{3}$ | 6.4.4435591023.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ |
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 *.