Normalized defining polynomial
\( x^{8} - 2x^{7} - 25x^{6} + 87x^{5} + 241x^{4} - 926x^{3} - 1354x^{2} + 2946x + 3867 \)
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: | \(-656988848944587\) \(\medspace = -\,3^{3}\cdot 2221^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(71.15\) | 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: | $3^{1/2}2221^{1/2}\approx 81.62720134857007$ | ||
Ramified primes: | \(3\), \(2221\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{15}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{3}a^{2}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{257925}a^{7}-\frac{1251}{85975}a^{6}-\frac{7474}{85975}a^{5}+\frac{7153}{85975}a^{4}-\frac{19868}{257925}a^{3}-\frac{5461}{85975}a^{2}-\frac{35607}{85975}a-\frac{42311}{85975}$
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: | $\frac{123}{17195}a^{7}+\frac{212}{10317}a^{6}-\frac{3267}{17195}a^{5}-\frac{339}{3439}a^{4}+\frac{56389}{17195}a^{3}+\frac{76093}{51585}a^{2}-\frac{57488}{3439}a-\frac{302331}{17195}$, $\frac{4288}{257925}a^{7}+\frac{552}{85975}a^{6}-\frac{31422}{85975}a^{5}+\frac{47769}{85975}a^{4}+\frac{1159306}{257925}a^{3}-\frac{203518}{85975}a^{2}-\frac{1813886}{85975}a-\frac{1415113}{85975}$, $\frac{8636}{257925}a^{7}-\frac{32723}{257925}a^{6}-\frac{47019}{85975}a^{5}+\frac{335573}{85975}a^{4}-\frac{214678}{257925}a^{3}-\frac{6244913}{257925}a^{2}+\frac{1010638}{85975}a+\frac{3727269}{85975}$, $\frac{15711}{85975}a^{7}-\frac{194329}{257925}a^{6}-\frac{256002}{85975}a^{5}+\frac{1908629}{85975}a^{4}-\frac{246043}{85975}a^{3}-\frac{42080239}{257925}a^{2}+\frac{8289649}{85975}a+\frac{28902142}{85975}$, $\frac{6349}{85975}a^{7}-\frac{29917}{85975}a^{6}-\frac{85848}{85975}a^{5}+\frac{798176}{85975}a^{4}-\frac{480872}{85975}a^{3}-\frac{5144417}{85975}a^{2}+\frac{4228631}{85975}a+\frac{10572568}{85975}$, $\frac{300282}{85975}a^{7}+\frac{3943517}{257925}a^{6}-\frac{3780119}{85975}a^{5}-\frac{20003817}{85975}a^{4}+\frac{851369}{85975}a^{3}+\frac{219058757}{257925}a^{2}+\frac{79103423}{85975}a+\frac{27646689}{85975}$ | 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: | \( 122178.072265 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{1}\cdot 122178.072265 \cdot 1}{2\cdot\sqrt{656988848944587}}\cr\approx \mathstrut & 0.958394065884 \end{aligned}\]
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
\(\Q(\sqrt{2221}) \) |
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 | ${\href{/padicField/2.8.0.1}{8} }$ | R | ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(2221\) | Deg $8$ | $2$ | $4$ | $4$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.6663.2t1.a.a | $1$ | $ 3 \cdot 2221 $ | \(\Q(\sqrt{-6663}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.2221.2t1.a.a | $1$ | $ 2221 $ | \(\Q(\sqrt{2221}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.6663.4t3.a.a | $2$ | $ 3 \cdot 2221 $ | 4.0.19989.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.19989.6t13.b.a | $4$ | $ 3^{2} \cdot 2221 $ | 6.0.59967.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.133186707.12t34.b.a | $4$ | $ 3^{3} \cdot 2221^{2}$ | 6.0.59967.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.98602558749.12t34.b.a | $4$ | $ 3^{2} \cdot 2221^{3}$ | 6.0.59967.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.14798523.6t13.b.a | $4$ | $ 3 \cdot 2221^{2}$ | 6.0.59967.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.98602558749.12t201.a.a | $6$ | $ 3^{2} \cdot 2221^{3}$ | 8.6.656988848944587.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
6.295807676247.12t202.a.a | $6$ | $ 3^{3} \cdot 2221^{3}$ | 8.6.656988848944587.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-4$ | |
* | 6.295807676247.8t47.a.a | $6$ | $ 3^{3} \cdot 2221^{3}$ | 8.6.656988848944587.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $4$ |
6.887423028741.12t200.a.a | $6$ | $ 3^{4} \cdot 2221^{3}$ | 8.6.656988848944587.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
9.266...223.16t1294.a.a | $9$ | $ 3^{5} \cdot 2221^{3}$ | 8.6.656988848944587.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $3$ | |
9.887423028741.18t272.a.a | $9$ | $ 3^{4} \cdot 2221^{3}$ | 8.6.656988848944587.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-3$ | |
9.972...001.18t273.a.a | $9$ | $ 3^{4} \cdot 2221^{6}$ | 8.6.656988848944587.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-3$ | |
9.291...003.18t274.a.a | $9$ | $ 3^{5} \cdot 2221^{6}$ | 8.6.656988848944587.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $3$ | |
12.262...027.36t1763.a.a | $12$ | $ 3^{7} \cdot 2221^{6}$ | 8.6.656988848944587.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.291...003.24t2821.a.a | $12$ | $ 3^{5} \cdot 2221^{6}$ | 8.6.656988848944587.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.258...223.36t1758.a.a | $18$ | $ 3^{9} \cdot 2221^{9}$ | 8.6.656988848944587.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |