Normalized defining polynomial
\( x^{6} - 9x^{4} - 132x^{3} + 464x^{2} - 116x - 614 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(2, 2)$ |
| |
| Discriminant: |
\(45812608000\)
\(\medspace = 2^{10}\cdot 5^{3}\cdot 71^{3}\)
|
| |
| Root discriminant: | \(59.82\) |
| |
| Galois root discriminant: | $2^{2}5^{5/6}71^{1/2}\approx 128.87366198354528$ | ||
| Ramified primes: |
\(2\), \(5\), \(71\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{355}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{68295}a^{5}-\frac{27367}{68295}a^{4}+\frac{5942}{13659}a^{3}-\frac{21727}{68295}a^{2}+\frac{9001}{22765}a+\frac{28978}{68295}$
| 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: |
$\frac{1294}{4553}a^{5}+\frac{9442}{4553}a^{4}+\frac{21973}{4553}a^{3}-\frac{159318}{4553}a^{2}-\frac{703555}{4553}a-\frac{506359}{4553}$, $\frac{9736}{68295}a^{5}+\frac{41978}{68295}a^{4}-\frac{144802}{13659}a^{3}+\frac{1751213}{68295}a^{2}-\frac{79809}{22765}a-\frac{2045687}{68295}$, $\frac{2372906281}{68295}a^{5}+\frac{7885689473}{68295}a^{4}+\frac{95360923487}{13659}a^{3}-\frac{1706420197507}{68295}a^{2}+\frac{178245909566}{22765}a+\frac{2370391497523}{68295}$
|
| |
| Regulator: | \( 8349.61544307 \) |
| |
| Unit signature rank: | \( 1 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 8349.61544307 \cdot 1}{2\cdot\sqrt{45812608000}}\cr\approx \mathstrut & 3.08009206702 \end{aligned}\]
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: | 6.2.14200000.1 |
| Degree 6 sibling: | 6.2.14200000.1 |
| 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: | 6.2.14200000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | ${\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
| 2.1.4.8b1.3 | $x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $D_{4}$ | $$[2, 3]^{2}$$ | |
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 5.1.2.1a1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 5.1.3.2a1.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
|
\(71\)
| 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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.1420.2t1.a.a | $1$ | $ 2^{2} \cdot 5 \cdot 71 $ | \(\Q(\sqrt{355}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *72 | 1.284.2t1.a.a | $1$ | $ 2^{2} \cdot 71 $ | \(\Q(\sqrt{71}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.22720.4t3.a.a | $2$ | $ 2^{6} \cdot 5 \cdot 71 $ | 4.0.113600.1 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
| 4.161312000.12t34.b.a | $4$ | $ 2^{8} \cdot 5^{3} \cdot 71^{2}$ | 6.2.45812608000.4 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| *72 | 4.161312000.6t13.b.a | $4$ | $ 2^{8} \cdot 5^{3} \cdot 71^{2}$ | 6.2.45812608000.4 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ |
| 4.2840000.6t13.a.a | $4$ | $ 2^{6} \cdot 5^{4} \cdot 71 $ | 6.2.45812608000.4 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.229063040000.12t34.a.a | $4$ | $ 2^{10} \cdot 5^{4} \cdot 71^{3}$ | 6.2.45812608000.4 | $C_3^2:D_4$ (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 *.