Normalized defining polynomial
\( x^{8} - 2x^{7} + x^{6} - 7x^{5} + 20x^{4} - 13x^{3} - 46x^{2} + 71x - 16 \)
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)
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Discriminant: | \(109049934277\) \(\medspace = 37\cdot 233^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(23.97\) | 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: | $37^{1/2}233^{1/2}\approx 92.84934033152847$ | ||
Ramified primes: | \(37\), \(233\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{37}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{9}a^{6}-\frac{1}{9}a^{5}+\frac{1}{9}a^{4}-\frac{1}{9}a^{3}+\frac{2}{9}a^{2}+\frac{1}{3}a+\frac{4}{9}$, $\frac{1}{63}a^{7}+\frac{1}{21}a^{6}+\frac{1}{7}a^{5}+\frac{1}{21}a^{4}+\frac{1}{9}a^{3}+\frac{8}{63}a^{2}+\frac{22}{63}a-\frac{8}{63}$
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: | $\frac{1}{21}a^{7}-\frac{5}{63}a^{6}-\frac{1}{63}a^{5}-\frac{26}{63}a^{4}+\frac{8}{9}a^{3}+\frac{17}{63}a^{2}-\frac{16}{7}a+\frac{109}{63}$, $\frac{5}{63}a^{7}-\frac{2}{21}a^{6}+\frac{1}{21}a^{5}-\frac{3}{7}a^{4}+\frac{11}{9}a^{3}-\frac{23}{63}a^{2}-\frac{289}{63}a+\frac{107}{63}$, $\frac{2}{63}a^{7}+\frac{2}{21}a^{6}-\frac{1}{21}a^{5}-\frac{5}{21}a^{4}-\frac{4}{9}a^{3}+\frac{79}{63}a^{2}-\frac{19}{63}a-\frac{247}{63}$, $\frac{16}{63}a^{7}-\frac{29}{63}a^{6}+\frac{11}{63}a^{5}-\frac{113}{63}a^{4}+\frac{14}{3}a^{3}-\frac{131}{63}a^{2}-\frac{761}{63}a+\frac{929}{63}$, $\frac{8}{63}a^{7}+\frac{17}{63}a^{6}-\frac{47}{63}a^{5}-\frac{130}{63}a^{4}-\frac{5}{3}a^{3}+\frac{596}{63}a^{2}+\frac{386}{63}a-\frac{869}{63}$ | 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: | \( 670.841566901 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 670.841566901 \cdot 1}{2\cdot\sqrt{109049934277}}\cr\approx \mathstrut & 0.641589110643 \end{aligned}\]
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
\(\Q(\sqrt{233}) \) |
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 | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
---|---|---|---|---|---|---|---|
\(37\) | $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(233\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $6$ | $2$ | $3$ | $3$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.37.2t1.a.a | $1$ | $ 37 $ | \(\Q(\sqrt{37}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.8621.2t1.a.a | $1$ | $ 37 \cdot 233 $ | \(\Q(\sqrt{8621}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.233.2t1.a.a | $1$ | $ 233 $ | \(\Q(\sqrt{233}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.8621.4t3.c.a | $2$ | $ 37 \cdot 233 $ | 4.0.318977.1 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
4.17316942353.12t34.b.a | $4$ | $ 37^{2} \cdot 233^{3}$ | 6.2.11802149.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.2749900717.12t34.b.a | $4$ | $ 37^{3} \cdot 233^{2}$ | 6.2.11802149.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.318977.6t13.b.a | $4$ | $ 37^{2} \cdot 233 $ | 6.2.11802149.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.2008693.6t13.b.a | $4$ | $ 37 \cdot 233^{2}$ | 6.2.11802149.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
6.237...257.12t201.a.a | $6$ | $ 37^{4} \cdot 233^{3}$ | 8.4.109049934277.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
6.877...509.12t202.a.a | $6$ | $ 37^{5} \cdot 233^{3}$ | 8.4.109049934277.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
* | 6.468025469.8t47.a.a | $6$ | $ 37 \cdot 233^{3}$ | 8.4.109049934277.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ |
6.17316942353.12t200.a.a | $6$ | $ 37^{2} \cdot 233^{3}$ | 8.4.109049934277.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.640726867061.16t1294.a.a | $9$ | $ 37^{3} \cdot 233^{3}$ | 8.4.109049934277.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.324...833.18t272.a.a | $9$ | $ 37^{6} \cdot 233^{3}$ | 8.4.109049934277.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.410...721.18t273.a.a | $9$ | $ 37^{6} \cdot 233^{6}$ | 8.4.109049934277.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.810...557.18t274.a.a | $9$ | $ 37^{3} \cdot 233^{6}$ | 8.4.109049934277.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
12.151...677.36t1763.a.a | $12$ | $ 37^{7} \cdot 233^{6}$ | 8.4.109049934277.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
12.110...533.24t2821.a.a | $12$ | $ 37^{5} \cdot 233^{6}$ | 8.4.109049934277.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
18.263...981.36t1758.a.a | $18$ | $ 37^{9} \cdot 233^{9}$ | 8.4.109049934277.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ |