Normalized defining polynomial
\( x^{8} - 4x^{7} + 2x^{6} - 2x^{5} + 37x^{4} - 42x^{3} - 62x^{2} + 116x - 44 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-94939073536\) \(\medspace = -\,2^{10}\cdot 19\cdot 47^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(23.56\) | 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: | $2^{3/2}19^{1/2}47^{1/2}\approx 84.52218643646175$ | ||
Ramified primes: | \(2\), \(19\), \(47\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{202}a^{7}+\frac{17}{101}a^{6}+\frac{41}{101}a^{5}+\frac{42}{101}a^{4}-\frac{3}{202}a^{3}+\frac{23}{101}a^{2}+\frac{35}{101}a-\frac{26}{101}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | 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: | $\frac{35}{101}a^{7}-\frac{145}{202}a^{6}-\frac{59}{101}a^{5}-\frac{191}{101}a^{4}+\frac{905}{101}a^{3}+\frac{291}{202}a^{2}-\frac{1792}{101}a+\frac{907}{101}$, $\frac{63}{202}a^{7}-\frac{181}{202}a^{6}-\frac{43}{101}a^{5}-\frac{81}{101}a^{4}+\frac{2033}{202}a^{3}-\frac{435}{202}a^{2}-\frac{2037}{101}a+\frac{1392}{101}$, $\frac{77}{202}a^{7}-\frac{311}{202}a^{6}+\frac{26}{101}a^{5}+\frac{2}{101}a^{4}+\frac{3203}{202}a^{3}-\frac{2417}{202}a^{2}-\frac{3466}{101}a+\frac{3351}{101}$, $\frac{14}{101}a^{7}-\frac{159}{202}a^{6}+\frac{37}{101}a^{5}+\frac{65}{101}a^{4}+\frac{766}{101}a^{3}-\frac{1641}{202}a^{2}-\frac{1747}{101}a+\frac{1999}{101}$ | 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: | \( 476.700228202 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 476.700228202 \cdot 1}{2\cdot\sqrt{94939073536}}\cr\approx \mathstrut & 0.767524886152 \end{aligned}\]
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
\(\Q(\sqrt{47}) \) |
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 | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.8 | $x^{4} + 2 x^{3} + 2$ | $4$ | $1$ | $6$ | $D_{4}$ | $[2, 2]^{2}$ |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
\(19\) | 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(47\) | 47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
47.6.3.1 | $x^{6} - 3102 x^{5} + 354590889 x^{4} + 6556431723382 x^{3} - 13099756575 x^{2} - 483399888 x - 4360566$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.19.2t1.a.a | $1$ | $ 19 $ | \(\Q(\sqrt{-19}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.3572.2t1.a.a | $1$ | $ 2^{2} \cdot 19 \cdot 47 $ | \(\Q(\sqrt{-893}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.188.2t1.a.a | $1$ | $ 2^{2} \cdot 47 $ | \(\Q(\sqrt{47}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.14288.4t3.b.a | $2$ | $ 2^{4} \cdot 19 \cdot 47 $ | 4.0.271472.2 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.271472.6t13.b.a | $4$ | $ 2^{4} \cdot 19^{2} \cdot 47 $ | 6.0.5157968.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.969697984.12t34.b.a | $4$ | $ 2^{6} \cdot 19^{3} \cdot 47^{2}$ | 6.0.5157968.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.9594906368.12t34.b.a | $4$ | $ 2^{8} \cdot 19^{2} \cdot 47^{3}$ | 6.0.5157968.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.2686144.6t13.b.a | $4$ | $ 2^{6} \cdot 19 \cdot 47^{2}$ | 6.0.5157968.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.138...392.12t201.a.a | $6$ | $ 2^{10} \cdot 19^{4} \cdot 47^{3}$ | 8.2.94939073536.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
6.658...112.12t202.a.a | $6$ | $ 2^{8} \cdot 19^{5} \cdot 47^{3}$ | 8.2.94939073536.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
* | 6.504995072.8t47.a.a | $6$ | $ 2^{8} \cdot 19 \cdot 47^{3}$ | 8.2.94939073536.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
6.38379625472.12t200.a.a | $6$ | $ 2^{10} \cdot 19^{2} \cdot 47^{3}$ | 8.2.94939073536.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.729212883968.16t1294.a.a | $9$ | $ 2^{10} \cdot 19^{3} \cdot 47^{3}$ | 8.2.94939073536.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
9.500...512.18t272.a.a | $9$ | $ 2^{10} \cdot 19^{6} \cdot 47^{3}$ | 8.2.94939073536.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.332...064.18t273.a.a | $9$ | $ 2^{16} \cdot 19^{6} \cdot 47^{6}$ | 8.2.94939073536.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.484...496.18t274.a.a | $9$ | $ 2^{16} \cdot 19^{3} \cdot 47^{6}$ | 8.2.94939073536.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
12.252...864.36t1763.a.a | $12$ | $ 2^{18} \cdot 19^{7} \cdot 47^{6}$ | 8.2.94939073536.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.699...224.24t2821.a.a | $12$ | $ 2^{18} \cdot 19^{5} \cdot 47^{6}$ | 8.2.94939073536.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.969...808.36t1758.a.a | $18$ | $ 2^{28} \cdot 19^{9} \cdot 47^{9}$ | 8.2.94939073536.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |