Normalized defining polynomial
\( x^{8} - 6x^{6} - 4x^{5} + 10x^{4} - 12x^{3} - 10x^{2} + 16x + 7 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(239251750912\) \(\medspace = 2^{18}\cdot 97^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(26.45\) | 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: | $2^{9/4}97^{1/2}\approx 46.84932709018395$ | ||
Ramified primes: | \(2\), \(97\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{97}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{749}a^{7}+\frac{10}{749}a^{6}+\frac{94}{749}a^{5}+\frac{187}{749}a^{4}-\frac{367}{749}a^{3}+\frac{9}{107}a^{2}-\frac{129}{749}a+\frac{32}{107}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $\frac{143}{749}a^{7}-\frac{68}{749}a^{6}-\frac{789}{749}a^{5}-\frac{223}{749}a^{4}+\frac{1447}{749}a^{3}-\frac{318}{107}a^{2}-\frac{471}{749}a+\frac{82}{107}$, $\frac{32}{107}a^{7}-\frac{1}{107}a^{6}-\frac{202}{107}a^{5}-\frac{115}{107}a^{4}+\frac{347}{107}a^{3}-\frac{445}{107}a^{2}-\frac{383}{107}a+\frac{748}{107}$, $\frac{12}{107}a^{7}+\frac{13}{107}a^{6}-\frac{49}{107}a^{5}-\frac{110}{107}a^{4}-\frac{17}{107}a^{3}-\frac{100}{107}a^{2}-\frac{264}{107}a-\frac{94}{107}$, $\frac{1005}{749}a^{7}-\frac{436}{749}a^{6}-\frac{5896}{749}a^{5}-\frac{1562}{749}a^{4}+\frac{10908}{749}a^{3}-\frac{2297}{107}a^{2}-\frac{2315}{749}a+\frac{2628}{107}$, $\frac{67}{107}a^{7}+\frac{28}{107}a^{6}-\frac{336}{107}a^{5}-\frac{418}{107}a^{4}+\frac{235}{107}a^{3}-\frac{808}{107}a^{2}-\frac{1046}{107}a+\frac{135}{107}$ | 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: | \( 1052.63865021 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 1052.63865021 \cdot 1}{2\cdot\sqrt{239251750912}}\cr\approx \mathstrut & 0.679675277980 \end{aligned}\]
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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.18.60 | $x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{5} + 2 x^{4} + 4 x^{3} + 4 x^{2} + 2$ | $8$ | $1$ | $18$ | $A_4\times C_2$ | $[2, 2, 3]^{3}$ |
\(97\) | 97.2.1.1 | $x^{2} + 97$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
97.2.1.1 | $x^{2} + 97$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
97.2.1.2 | $x^{2} + 485$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.97.2t1.a.a | $1$ | $ 97 $ | \(\Q(\sqrt{97}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.776.2t1.a.a | $1$ | $ 2^{3} \cdot 97 $ | \(\Q(\sqrt{194}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.776.4t3.b.a | $2$ | $ 2^{3} \cdot 97 $ | 4.0.75272.1 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
4.75272.6t13.a.a | $4$ | $ 2^{3} \cdot 97^{2}$ | 6.2.7301384.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.58411072.12t34.a.a | $4$ | $ 2^{6} \cdot 97^{3}$ | 6.2.7301384.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.4817408.12t34.a.a | $4$ | $ 2^{9} \cdot 97^{2}$ | 6.2.7301384.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.6208.6t13.a.a | $4$ | $ 2^{6} \cdot 97 $ | 6.2.7301384.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
6.308314112.12t201.a.a | $6$ | $ 2^{15} \cdot 97^{2}$ | 8.4.239251750912.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
6.29906468864.12t202.b.a | $6$ | $ 2^{15} \cdot 97^{3}$ | 8.4.239251750912.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
* | 6.29906468864.8t47.a.a | $6$ | $ 2^{15} \cdot 97^{3}$ | 8.4.239251750912.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ |
6.290...808.12t200.a.a | $6$ | $ 2^{15} \cdot 97^{4}$ | 8.4.239251750912.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.281...376.16t1294.a.a | $9$ | $ 2^{15} \cdot 97^{5}$ | 8.4.239251750912.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.290...808.18t272.a.a | $9$ | $ 2^{15} \cdot 97^{4}$ | 8.4.239251750912.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.148...696.18t273.a.a | $9$ | $ 2^{24} \cdot 97^{4}$ | 8.4.239251750912.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.144...512.18t274.a.a | $9$ | $ 2^{24} \cdot 97^{5}$ | 8.4.239251750912.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
12.867...112.36t1763.a.a | $12$ | $ 2^{30} \cdot 97^{7}$ | 8.4.239251750912.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
12.922...768.24t2821.a.a | $12$ | $ 2^{30} \cdot 97^{5}$ | 8.4.239251750912.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
18.417...696.36t1758.a.a | $18$ | $ 2^{39} \cdot 97^{9}$ | 8.4.239251750912.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ |