Normalized defining polynomial
\( x^{18} + 5x^{16} + 5x^{14} - 8x^{12} - 8x^{10} + 17x^{8} + 19x^{6} + x^{4} - 2x^{2} - 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(15335719659532189696\) \(\medspace = 2^{18}\cdot 37^{6}\cdot 151^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(11.64\) | 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))
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Galois root discriminant: | not computed | ||
Ramified primes: | \(2\), \(37\), \(151\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{12}-\frac{1}{4}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a-\frac{1}{4}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $\frac{3}{4}a^{17}+\frac{1}{4}a^{16}+3a^{15}+\frac{5}{4}a^{14}+\frac{5}{4}a^{13}+\frac{5}{4}a^{12}-\frac{11}{2}a^{11}-\frac{7}{4}a^{10}-\frac{1}{2}a^{9}-\frac{3}{2}a^{8}+\frac{19}{2}a^{7}+\frac{7}{2}a^{6}+\frac{23}{4}a^{5}+\frac{17}{4}a^{4}+\frac{1}{2}a^{3}+\frac{7}{4}a^{2}-a-\frac{1}{2}$, $\frac{1}{2}a^{16}+\frac{3}{2}a^{14}-\frac{1}{2}a^{12}-\frac{5}{2}a^{10}+3a^{8}+3a^{6}-\frac{1}{2}a^{4}+\frac{5}{2}a^{2}$, $\frac{1}{4}a^{17}+\frac{1}{4}a^{16}+\frac{3}{2}a^{15}+\frac{3}{4}a^{14}+\frac{7}{4}a^{13}-\frac{1}{4}a^{12}-3a^{11}-\frac{5}{4}a^{10}-3a^{9}+\frac{3}{2}a^{8}+7a^{7}+\frac{3}{2}a^{6}+\frac{21}{4}a^{5}-\frac{1}{4}a^{4}-\frac{3}{2}a^{3}+\frac{7}{4}a^{2}+\frac{1}{2}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{16}+\frac{3}{2}a^{15}-\frac{3}{4}a^{14}+\frac{7}{4}a^{13}+\frac{1}{4}a^{12}-3a^{11}+\frac{5}{4}a^{10}-3a^{9}-\frac{3}{2}a^{8}+7a^{7}-\frac{3}{2}a^{6}+\frac{21}{4}a^{5}+\frac{1}{4}a^{4}-\frac{3}{2}a^{3}-\frac{7}{4}a^{2}-\frac{1}{2}$, $a$, $\frac{1}{4}a^{17}+a^{16}+\frac{5}{4}a^{15}+\frac{15}{4}a^{14}+a^{13}+\frac{1}{2}a^{12}-\frac{11}{4}a^{11}-\frac{31}{4}a^{10}-\frac{7}{4}a^{9}+\frac{5}{2}a^{8}+6a^{7}+13a^{6}+\frac{15}{4}a^{5}+\frac{5}{2}a^{4}-\frac{11}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{4}a^{17}-a^{16}+\frac{5}{4}a^{15}-\frac{15}{4}a^{14}+a^{13}-\frac{1}{2}a^{12}-\frac{11}{4}a^{11}+\frac{31}{4}a^{10}-\frac{7}{4}a^{9}-\frac{5}{2}a^{8}+6a^{7}-13a^{6}+\frac{15}{4}a^{5}-\frac{5}{2}a^{4}-\frac{11}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{4}a^{17}-\frac{3}{4}a^{16}+\frac{1}{2}a^{15}-\frac{11}{4}a^{14}-\frac{5}{4}a^{13}-\frac{1}{4}a^{12}-\frac{3}{2}a^{11}+\frac{23}{4}a^{10}+\frac{7}{2}a^{9}-\frac{3}{2}a^{8}+a^{7}-9a^{6}-\frac{13}{4}a^{5}-\frac{11}{4}a^{4}+\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-a+\frac{3}{2}$, $\frac{1}{4}a^{17}+\frac{3}{4}a^{16}+\frac{1}{2}a^{15}+\frac{11}{4}a^{14}-\frac{5}{4}a^{13}+\frac{1}{4}a^{12}-\frac{3}{2}a^{11}-\frac{23}{4}a^{10}+\frac{7}{2}a^{9}+\frac{3}{2}a^{8}+a^{7}+9a^{6}-\frac{13}{4}a^{5}+\frac{11}{4}a^{4}+\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-a-\frac{3}{2}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 139.095050977 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 139.095050977 \cdot 1}{2\cdot\sqrt{15335719659532189696}}\cr\approx \mathstrut & 0.172555394152 \end{aligned}\]
Galois group
$S_3^3:S_4$ (as 18T483):
A solvable group of order 5184 |
The 49 conjugacy class representatives for $S_3^3:S_4$ |
Character table for $S_3^3:S_4$ |
Intermediate fields
3.3.148.1, 6.2.87616.1, 9.3.489510592.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 18 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 | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.9.0.1}{9} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.18.18.124 | $x^{18} + 6 x^{14} + 6 x^{13} + 6 x^{12} + 16 x^{10} + 24 x^{9} + 36 x^{8} + 24 x^{7} + 20 x^{6} + 32 x^{5} + 56 x^{4} + 56 x^{3} + 48 x^{2} + 24 x + 8$ | $6$ | $3$ | $18$ | 18T33 | $[4/3, 4/3]_{3}^{6}$ |
\(37\) | 37.3.0.1 | $x^{3} + 6 x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
37.3.0.1 | $x^{3} + 6 x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(151\) | $\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |