Normalized defining polynomial
\( x^{18} - 5 x^{17} + 18 x^{16} - 49 x^{15} + 107 x^{14} - 195 x^{13} + 299 x^{12} - 386 x^{11} + 419 x^{10} + \cdots + 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-33019305672452059136\) \(\medspace = -\,2^{12}\cdot 11^{9}\cdot 43^{4}\) | 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: | \(12.14\) | 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: | $2^{2/3}11^{1/2}43^{2/3}\approx 64.61925947014737$ | ||
Ramified primes: | \(2\), \(11\), \(43\) | 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: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{6349}a^{17}+\frac{991}{6349}a^{16}+\frac{2959}{6349}a^{15}+\frac{1179}{6349}a^{14}-\frac{174}{6349}a^{13}-\frac{2076}{6349}a^{12}+\frac{2377}{6349}a^{11}-\frac{153}{907}a^{10}+\frac{335}{6349}a^{9}+\frac{3135}{6349}a^{8}-\frac{972}{6349}a^{7}+\frac{3121}{6349}a^{6}-\frac{2410}{6349}a^{5}-\frac{69}{907}a^{4}+\frac{1486}{6349}a^{3}+\frac{719}{6349}a^{2}-\frac{186}{907}a-\frac{1600}{6349}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $\frac{7459}{6349}a^{17}-\frac{30112}{6349}a^{16}+\frac{103641}{6349}a^{15}-\frac{259513}{6349}a^{14}+\frac{524297}{6349}a^{13}-\frac{888533}{6349}a^{12}+\frac{1241690}{6349}a^{11}-\frac{206110}{907}a^{10}+\frac{1374992}{6349}a^{9}-\frac{1027940}{6349}a^{8}+\frac{584518}{6349}a^{7}-\frac{256204}{6349}a^{6}+\frac{118460}{6349}a^{5}-\frac{15821}{907}a^{4}+\frac{100304}{6349}a^{3}-\frac{46327}{6349}a^{2}+\frac{2150}{907}a+\frac{1720}{6349}$, $\frac{11020}{6349}a^{17}-\frac{50252}{6349}a^{16}+\frac{171139}{6349}a^{15}-\frac{441904}{6349}a^{14}+\frac{907825}{6349}a^{13}-\frac{1551229}{6349}a^{12}+\frac{2208018}{6349}a^{11}-\frac{370003}{907}a^{10}+\frac{2491739}{6349}a^{9}-\frac{1895560}{6349}a^{8}+\frac{1072304}{6349}a^{7}-\frac{449892}{6349}a^{6}+\frac{196486}{6349}a^{5}-\frac{22082}{907}a^{4}+\frac{141327}{6349}a^{3}-\frac{76360}{6349}a^{2}+\frac{2821}{907}a-\frac{827}{6349}$, $\frac{1520}{6349}a^{17}-\frac{11091}{6349}a^{16}+\frac{40682}{6349}a^{15}-\frac{118969}{6349}a^{14}+\frac{268836}{6349}a^{13}-\frac{501638}{6349}a^{12}+\frac{781386}{6349}a^{11}-\frac{144581}{907}a^{10}+\frac{1080610}{6349}a^{9}-\frac{936202}{6349}a^{8}+\frac{624079}{6349}a^{7}-\frac{309884}{6349}a^{6}+\frac{127153}{6349}a^{5}-\frac{11459}{907}a^{4}+\frac{81013}{6349}a^{3}-\frac{56289}{6349}a^{2}+\frac{2985}{907}a-\frac{6682}{6349}$, $a$, $\frac{12521}{6349}a^{17}-\frac{54776}{6349}a^{16}+\frac{187345}{6349}a^{15}-\frac{481690}{6349}a^{14}+\frac{989497}{6349}a^{13}-\frac{1702322}{6349}a^{12}+\frac{2442670}{6349}a^{11}-\frac{416442}{907}a^{10}+\frac{2892990}{6349}a^{9}-\frac{2319917}{6349}a^{8}+\frac{1467240}{6349}a^{7}-\frac{761934}{6349}a^{6}+\frac{388476}{6349}a^{5}-\frac{35858}{907}a^{4}+\frac{181408}{6349}a^{3}-\frac{95518}{6349}a^{2}+\frac{4805}{907}a-\frac{2505}{6349}$, $\frac{1633}{6349}a^{17}-\frac{13390}{6349}a^{16}+\frac{51250}{6349}a^{15}-\frac{150816}{6349}a^{14}+\frac{350758}{6349}a^{13}-\frac{660038}{6349}a^{12}+\frac{1043638}{6349}a^{11}-\frac{196336}{907}a^{10}+\frac{1486707}{6349}a^{9}-\frac{1318431}{6349}a^{8}+\frac{914230}{6349}a^{7}-\frac{477829}{6349}a^{6}+\frac{223065}{6349}a^{5}-\frac{20163}{907}a^{4}+\frac{115602}{6349}a^{3}-\frac{76626}{6349}a^{2}+\frac{3735}{907}a-\frac{3361}{6349}$, $\frac{2273}{907}a^{17}-\frac{9515}{907}a^{16}+\frac{32147}{907}a^{15}-\frac{81041}{907}a^{14}+\frac{163210}{907}a^{13}-\frac{275355}{907}a^{12}+\frac{385397}{907}a^{11}-\frac{446239}{907}a^{10}+\frac{425865}{907}a^{9}-\frac{322429}{907}a^{8}+\frac{186938}{907}a^{7}-\frac{86686}{907}a^{6}+\frac{43886}{907}a^{5}-\frac{32134}{907}a^{4}+\frac{26313}{907}a^{3}-\frac{14639}{907}a^{2}+\frac{5537}{907}a-\frac{637}{907}$, $\frac{8816}{6349}a^{17}-\frac{44011}{6349}a^{16}+\frac{157228}{6349}a^{15}-\frac{424632}{6349}a^{14}+\frac{916730}{6349}a^{13}-\frac{1648589}{6349}a^{12}+\frac{2480042}{6349}a^{11}-\frac{446383}{907}a^{10}+\frac{3270810}{6349}a^{9}-\frac{2792597}{6349}a^{8}+\frac{1881302}{6349}a^{7}-\frac{985925}{6349}a^{6}+\frac{447973}{6349}a^{5}-\frac{37801}{907}a^{4}+\frac{218455}{6349}a^{3}-\frac{143625}{6349}a^{2}+\frac{8243}{907}a-\frac{10820}{6349}$ | 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: | \( 134.134877065 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 134.134877065 \cdot 1}{2\cdot\sqrt{33019305672452059136}}\cr\approx \mathstrut & 0.178133896093 \end{aligned}\]
Galois group
$C_3\wr S_3$ (as 18T86):
A solvable group of order 162 |
The 22 conjugacy class representatives for $C_3\wr S_3$ |
Character table for $C_3\wr S_3$ |
Intermediate fields
\(\Q(\sqrt{-11}) \), 3.1.44.1 x3, 6.0.21296.1, 9.3.157505216.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 9 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 27 siblings: | data not computed |
Minimal sibling: | 9.3.157505216.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.9.0.1}{9} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | R | ${\href{/padicField/47.3.0.1}{3} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{6}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.18.12.1 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
\(11\) | 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(43\) | 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
43.6.4.1 | $x^{6} + 126 x^{5} + 5301 x^{4} + 74930 x^{3} + 21321 x^{2} + 227916 x + 3171406$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |