Normalized defining polynomial
\( x^{6} - 12x^{4} - 56x^{3} + 36x^{2} + 336x - 352 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(2, 2)$ |
| |
| Discriminant: |
\(6597015552\)
\(\medspace = 2^{11}\cdot 3^{2}\cdot 71^{3}\)
|
| |
| Root discriminant: | \(43.31\) |
| |
| Galois root discriminant: | $2^{11/4}3^{1/2}71^{1/2}\approx 98.17983316921564$ | ||
| Ramified primes: |
\(2\), \(3\), \(71\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{142}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{24}a^{4}+\frac{1}{12}a^{3}+\frac{1}{12}a^{2}+\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{24}a^{5}-\frac{1}{12}a^{3}+\frac{1}{3}a-\frac{1}{3}$
| 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: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{19}{24}a^{5}+\frac{29}{24}a^{4}-\frac{23}{3}a^{3}-\frac{673}{12}a^{2}-\frac{341}{6}a+181$, $\frac{17}{12}a^{5}+\frac{13}{4}a^{4}-\frac{73}{12}a^{3}-\frac{147}{2}a^{2}-\frac{397}{6}a+\frac{941}{3}$, $\frac{41}{12}a^{5}-\frac{179}{8}a^{4}+\frac{331}{6}a^{3}+\frac{615}{4}a^{2}-\frac{1745}{3}a+\frac{1295}{3}$
|
| |
| Regulator: | \( 1890.63263646 \) |
| |
| Unit signature rank: | \( 1 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 1890.63263646 \cdot 1}{2\cdot\sqrt{6597015552}}\cr\approx \mathstrut & 1.83790529483 \end{aligned}\]
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
| \(\Q(\sqrt{71}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 6.2.5234688.1 |
| Degree 6 sibling: | 6.2.5234688.1 |
| 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: | 6.2.5234688.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
| 2.1.4.9a1.1 | $x^{4} + 2 x^{2} + 2$ | $4$ | $1$ | $9$ | $D_{4}$ | $$[2, 3, \frac{7}{2}]$$ | |
|
\(3\)
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 3.2.2.2a1.1 | $x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ | |
|
\(71\)
| 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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.568.2t1.a.a | $1$ | $ 2^{3} \cdot 71 $ | \(\Q(\sqrt{142}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *72 | 1.284.2t1.a.a | $1$ | $ 2^{2} \cdot 71 $ | \(\Q(\sqrt{71}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.81792.4t3.b.a | $2$ | $ 2^{7} \cdot 3^{2} \cdot 71 $ | 4.0.654336.1 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
| 4.371662848.12t34.a.a | $4$ | $ 2^{13} \cdot 3^{2} \cdot 71^{2}$ | 6.2.6597015552.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| *72 | 4.23228928.6t13.a.a | $4$ | $ 2^{9} \cdot 3^{2} \cdot 71^{2}$ | 6.2.6597015552.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ |
| 4.654336.6t13.a.a | $4$ | $ 2^{10} \cdot 3^{2} \cdot 71 $ | 6.2.6597015552.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.13194031104.12t34.a.a | $4$ | $ 2^{12} \cdot 3^{2} \cdot 71^{3}$ | 6.2.6597015552.1 | $C_3^2:D_4$ (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 *.