Normalized defining polynomial
\( x^{6} - 2x^{5} - x^{4} - 31x^{3} + 34x^{2} + 33x + 6 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(4, 1)$ |
| |
| Discriminant: |
\(-42278236875\)
\(\medspace = -\,3^{3}\cdot 5^{4}\cdot 7\cdot 71^{3}\)
|
| |
| Root discriminant: | \(59.02\) |
| |
| Galois root discriminant: | $3^{1/2}5^{3/4}7^{1/2}71^{1/2}\approx 129.11191568658924$ | ||
| Ramified primes: |
\(3\), \(5\), \(7\), \(71\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-1491}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{2}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{2}\times C_{2}\times C_{2}$, which has order $8$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{3}{4}a^{5}-\frac{7}{4}a^{4}-\frac{1}{2}a^{3}-\frac{87}{4}a^{2}+\frac{131}{4}a+\frac{25}{2}$, $6a^{5}-\frac{29}{2}a^{4}-186a^{2}+\frac{563}{2}a+82$, $\frac{67}{4}a^{5}-\frac{157}{4}a^{4}-\frac{5}{2}a^{3}-\frac{2075}{4}a^{2}+\frac{2985}{4}a+\frac{539}{2}$, $629a^{5}-\frac{2959}{2}a^{4}-175a^{3}-19395a^{2}+\frac{56615}{2}a+13012$
|
| |
| Regulator: | \( 4590.78581543 \) |
| |
| Unit signature rank: | \( 2 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{1}\cdot 4590.78581543 \cdot 2}{2\cdot\sqrt{42278236875}}\cr\approx \mathstrut & 2.24454567934 \end{aligned}\]
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
| \(\Q(\sqrt{1065}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 6.0.45661875.1 |
| Degree 6 sibling: | 6.0.45661875.1 |
| 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: | 6.0.45661875.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | R | R | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(5\)
| 5.1.2.1a1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.1.4.3a1.2 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 7.1.2.1a1.2 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 7.3.1.0a1.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
|
\(71\)
| 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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.1491.2t1.a.a | $1$ | $ 3 \cdot 7 \cdot 71 $ | \(\Q(\sqrt{-1491}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *72 | 1.1065.2t1.a.a | $1$ | $ 3 \cdot 5 \cdot 71 $ | \(\Q(\sqrt{1065}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.35.2t1.a.a | $1$ | $ 5 \cdot 7 $ | \(\Q(\sqrt{-35}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.37275.4t3.c.a | $2$ | $ 3 \cdot 5^{2} \cdot 7 \cdot 71 $ | 4.0.1304625.3 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| 4.1945195875.12t34.a.a | $4$ | $ 3^{2} \cdot 5^{3} \cdot 7^{3} \cdot 71^{2}$ | 6.4.42278236875.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
| *72 | 4.39697875.6t13.a.a | $4$ | $ 3^{2} \cdot 5^{3} \cdot 7 \cdot 71^{2}$ | 6.4.42278236875.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ |
| 4.1304625.6t13.a.a | $4$ | $ 3 \cdot 5^{3} \cdot 7^{2} \cdot 71 $ | 6.4.42278236875.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.59189531625.12t34.a.a | $4$ | $ 3^{3} \cdot 5^{3} \cdot 7^{2} \cdot 71^{3}$ | 6.4.42278236875.1 | $C_3^2:D_4$ (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 *.