Normalized defining polynomial
\( x^{8} - 12x^{5} - 30x^{4} + 32x^{3} + 28x^{2} - 12x + 1 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[6, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-82278203392\) \(\medspace = -\,2^{14}\cdot 7^{3}\cdot 11^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(23.14\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2^{11/6}7^{1/2}11^{2/3}\approx 46.633620646062376$ | ||
Ramified primes: | \(2\), \(7\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{124}a^{7}-\frac{15}{124}a^{6}-\frac{23}{124}a^{5}+\frac{23}{124}a^{4}-\frac{3}{124}a^{3}-\frac{47}{124}a^{2}-\frac{11}{124}a-\frac{33}{124}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{47}{62}a^{7}+\frac{4}{31}a^{6}+\frac{2}{31}a^{5}-\frac{281}{31}a^{4}-\frac{1505}{62}a^{3}+\frac{616}{31}a^{2}+\frac{718}{31}a-\frac{109}{31}$, $a$, $\frac{1}{124}a^{7}+\frac{4}{31}a^{6}-\frac{23}{124}a^{5}-\frac{2}{31}a^{4}-\frac{251}{124}a^{3}-\frac{101}{62}a^{2}+\frac{1105}{124}a-\frac{125}{62}$, $\frac{127}{124}a^{7}+\frac{17}{124}a^{6}-\frac{7}{124}a^{5}-\frac{1543}{124}a^{4}-\frac{4039}{124}a^{3}+\frac{3579}{124}a^{2}+\frac{4431}{124}a-\frac{657}{124}$, $\frac{103}{31}a^{7}+\frac{51}{124}a^{6}+\frac{5}{62}a^{5}-\frac{4939}{124}a^{4}-\frac{6477}{62}a^{3}+\frac{11543}{124}a^{2}+\frac{3238}{31}a-\frac{3335}{124}$, $\frac{27}{124}a^{7}-\frac{1}{62}a^{6}-\frac{1}{124}a^{5}-\frac{85}{31}a^{4}-\frac{825}{124}a^{3}+\frac{202}{31}a^{2}+\frac{695}{124}a-\frac{27}{62}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 1214.43857717 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{1}\cdot 1214.43857717 \cdot 1}{2\cdot\sqrt{82278203392}}\cr\approx \mathstrut & 0.851261517669 \end{aligned}\]
Galois group
$S_4\wr C_2$ (as 8T47):
A solvable group of order 1152 |
The 20 conjugacy class representatives for $S_4\wr C_2$ |
Character table for $S_4\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{2}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Degree 36 siblings: | data not computed |
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 | R | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.8.0.1}{8} }$ | R | R | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.6.11.1 | $x^{6} + 4 x^{3} + 2$ | $6$ | $1$ | $11$ | $D_{6}$ | $[3]_{3}^{2}$ | |
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.4.2.2 | $x^{4} - 42 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(11\) | 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
11.6.4.2 | $x^{6} - 110 x^{3} - 16819$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.56.2t1.b.a | $1$ | $ 2^{3} \cdot 7 $ | \(\Q(\sqrt{-14}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.56.4t3.b.a | $2$ | $ 2^{3} \cdot 7 $ | 4.0.392.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.3035648.12t34.a.a | $4$ | $ 2^{9} \cdot 7^{2} \cdot 11^{2}$ | 6.0.1328096.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.1285596928.12t34.i.a | $4$ | $ 2^{8} \cdot 7^{3} \cdot 11^{4}$ | 6.0.1328096.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.189728.6t13.a.a | $4$ | $ 2^{5} \cdot 7^{2} \cdot 11^{2}$ | 6.0.1328096.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.26236672.6t13.a.a | $4$ | $ 2^{8} \cdot 7 \cdot 11^{4}$ | 6.0.1328096.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.1469253632.12t201.a.a | $6$ | $ 2^{11} \cdot 7^{2} \cdot 11^{4}$ | 8.6.82278203392.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
6.10284775424.12t202.a.a | $6$ | $ 2^{11} \cdot 7^{3} \cdot 11^{4}$ | 8.6.82278203392.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-4$ | |
* | 6.10284775424.8t47.a.a | $6$ | $ 2^{11} \cdot 7^{3} \cdot 11^{4}$ | 8.6.82278203392.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $4$ |
6.71993427968.12t200.a.a | $6$ | $ 2^{11} \cdot 7^{4} \cdot 11^{4}$ | 8.6.82278203392.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
9.243...584.16t1294.a.a | $9$ | $ 2^{13} \cdot 7^{5} \cdot 11^{6}$ | 8.6.82278203392.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $3$ | |
9.348...512.18t272.a.a | $9$ | $ 2^{13} \cdot 7^{4} \cdot 11^{6}$ | 8.6.82278203392.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-3$ | |
9.446...536.18t273.b.a | $9$ | $ 2^{20} \cdot 7^{4} \cdot 11^{6}$ | 8.6.82278203392.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-3$ | |
9.312...752.18t274.a.a | $9$ | $ 2^{20} \cdot 7^{5} \cdot 11^{6}$ | 8.6.82278203392.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $3$ | |
12.740...432.36t1763.a.a | $12$ | $ 2^{22} \cdot 7^{7} \cdot 11^{8}$ | 8.6.82278203392.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.151...968.24t2821.a.a | $12$ | $ 2^{22} \cdot 7^{5} \cdot 11^{8}$ | 8.6.82278203392.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.108...024.36t1758.a.a | $18$ | $ 2^{33} \cdot 7^{9} \cdot 11^{12}$ | 8.6.82278203392.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |