Normalized defining polynomial
\( x^{18} - 30x^{15} + 360x^{12} - 1672x^{9} + 6720x^{6} - 6048x^{3} + 21952 \)
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: | \(-520986863358984384396363313152\) \(\medspace = -\,2^{12}\cdot 3^{37}\cdot 7^{10}\) | 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: | \(44.76\) | 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: | $2^{2/3}3^{37/18}7^{5/6}\approx 76.8571500447917$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | 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: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}$, $\frac{1}{4}a^{7}$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{24}a^{9}-\frac{1}{3}$, $\frac{1}{24}a^{10}-\frac{1}{3}a$, $\frac{1}{48}a^{11}-\frac{1}{6}a^{2}$, $\frac{1}{336}a^{12}+\frac{1}{14}a^{6}-\frac{1}{6}a^{3}$, $\frac{1}{672}a^{13}-\frac{5}{56}a^{7}+\frac{1}{6}a^{4}-\frac{1}{2}a$, $\frac{1}{2016}a^{14}-\frac{1}{2016}a^{13}-\frac{1}{1008}a^{12}-\frac{1}{144}a^{11}-\frac{1}{72}a^{10}+\frac{1}{72}a^{9}-\frac{5}{168}a^{8}+\frac{5}{168}a^{7}+\frac{5}{84}a^{6}-\frac{1}{9}a^{5}+\frac{1}{9}a^{4}+\frac{2}{9}a^{3}-\frac{1}{9}a^{2}+\frac{5}{18}a+\frac{2}{9}$, $\frac{1}{5162976}a^{15}+\frac{1423}{2581488}a^{12}+\frac{4513}{322686}a^{9}-\frac{71437}{645372}a^{6}-\frac{1403}{23049}a^{3}-\frac{1480}{23049}$, $\frac{1}{10325952}a^{16}+\frac{1423}{5162976}a^{13}-\frac{35729}{2581488}a^{10}-\frac{71437}{1290744}a^{7}+\frac{20243}{92196}a^{4}+\frac{6203}{46098}a$, $\frac{1}{72281664}a^{17}-\frac{8821}{36140832}a^{14}+\frac{1}{2016}a^{13}+\frac{1}{1008}a^{12}+\frac{1870}{376467}a^{11}+\frac{1}{72}a^{10}-\frac{1}{72}a^{9}+\frac{404909}{9035208}a^{8}-\frac{5}{168}a^{7}-\frac{5}{84}a^{6}+\frac{61219}{645372}a^{5}-\frac{1}{9}a^{4}-\frac{2}{9}a^{3}-\frac{6627}{17927}a^{2}-\frac{5}{18}a-\frac{2}{9}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{1}{15366} a^{15} + \frac{281}{122928} a^{12} - \frac{250}{7683} a^{9} + \frac{1477}{7683} a^{6} - \frac{4207}{7683} a^{3} + \frac{9539}{7683} \) (order $6$) | 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: | $\frac{3817}{36140832}a^{17}+\frac{3127}{10325952}a^{16}-\frac{977}{5162976}a^{15}-\frac{131191}{36140832}a^{14}-\frac{22417}{2581488}a^{13}+\frac{7859}{1290744}a^{12}+\frac{936323}{18070416}a^{11}+\frac{247459}{2581488}a^{10}-\frac{52099}{645372}a^{9}-\frac{1526593}{4517604}a^{8}-\frac{215261}{645372}a^{7}+\frac{150607}{322686}a^{6}+\frac{867379}{645372}a^{5}+\frac{99503}{92196}a^{4}-\frac{93565}{46098}a^{3}-\frac{780293}{322686}a^{2}+\frac{29335}{23049}a+\frac{93752}{23049}$, $\frac{3817}{36140832}a^{17}+\frac{3127}{10325952}a^{16}-\frac{977}{5162976}a^{15}-\frac{131191}{36140832}a^{14}-\frac{22417}{2581488}a^{13}+\frac{7859}{1290744}a^{12}+\frac{936323}{18070416}a^{11}+\frac{247459}{2581488}a^{10}-\frac{52099}{645372}a^{9}-\frac{1526593}{4517604}a^{8}-\frac{215261}{645372}a^{7}+\frac{150607}{322686}a^{6}+\frac{867379}{645372}a^{5}+\frac{99503}{92196}a^{4}-\frac{93565}{46098}a^{3}-\frac{780293}{322686}a^{2}+\frac{29335}{23049}a+\frac{70703}{23049}$, $\frac{547}{9035208}a^{17}-\frac{307}{2581488}a^{15}-\frac{28823}{18070416}a^{14}+\frac{9823}{2581488}a^{12}+\frac{32521}{2258802}a^{11}-\frac{58823}{1290744}a^{9}-\frac{40823}{9035208}a^{8}+\frac{28829}{161343}a^{6}-\frac{24671}{645372}a^{5}-\frac{6737}{23049}a^{3}+\frac{92485}{322686}a^{2}+\frac{25175}{23049}$, $\frac{215}{36140832}a^{17}-\frac{439}{5162976}a^{15}-\frac{10433}{36140832}a^{14}+\frac{5309}{2581488}a^{12}+\frac{71677}{18070416}a^{11}-\frac{19021}{1290744}a^{9}-\frac{85789}{9035208}a^{8}-\frac{31895}{645372}a^{6}-\frac{25057}{161343}a^{5}+\frac{1277}{23049}a^{3}-\frac{10880}{161343}a^{2}+\frac{4348}{23049}$, $\frac{1973}{36140832}a^{17}-\frac{25}{737568}a^{15}-\frac{47213}{36140832}a^{14}+\frac{2257}{1290744}a^{12}+\frac{188491}{18070416}a^{11}-\frac{2843}{92196}a^{9}+\frac{22483}{4517604}a^{8}+\frac{147211}{645372}a^{6}+\frac{75557}{645372}a^{5}-\frac{8014}{23049}a^{3}+\frac{114245}{322686}a^{2}+\frac{20827}{23049}$, $\frac{1}{44128}a^{15}-\frac{65}{66192}a^{12}+\frac{125}{11032}a^{9}-\frac{123}{5516}a^{6}+\frac{53}{1182}a^{3}+\frac{96}{197}$, $\frac{44801}{72281664}a^{17}+\frac{83}{61464}a^{16}-\frac{179}{107562}a^{15}-\frac{388847}{18070416}a^{14}-\frac{9977}{286832}a^{13}+\frac{23869}{430248}a^{12}+\frac{5313293}{18070416}a^{11}+\frac{18451}{61464}a^{10}-\frac{304219}{430248}a^{9}-\frac{1755004}{1129401}a^{8}-\frac{13007}{215124}a^{7}+\frac{711797}{215124}a^{6}+\frac{1741681}{645372}a^{5}+\frac{3989}{5122}a^{4}-\frac{33310}{7683}a^{3}-\frac{692065}{161343}a^{2}+\frac{30635}{7683}a+\frac{84778}{7683}$, $\frac{44801}{72281664}a^{17}-\frac{1619}{1720992}a^{16}+\frac{2075}{860496}a^{15}-\frac{388847}{18070416}a^{14}+\frac{1507}{53781}a^{13}-\frac{59119}{860496}a^{12}+\frac{5313293}{18070416}a^{11}-\frac{130667}{430248}a^{10}+\frac{151813}{215124}a^{9}-\frac{1755004}{1129401}a^{8}+\frac{86665}{107562}a^{7}-\frac{368413}{215124}a^{6}+\frac{1741681}{645372}a^{5}+\frac{4972}{7683}a^{4}+\frac{66553}{15366}a^{3}-\frac{692065}{161343}a^{2}+\frac{14390}{7683}a-\frac{62186}{7683}$ (assuming GRH) | 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: | \( 36855368.88515014 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 36855368.88515014 \cdot 3}{6\cdot\sqrt{520986863358984384396363313152}}\cr\approx \mathstrut & 0.389651359284569 \end{aligned}\] (assuming GRH)
Galois group
$C_3\times S_3^2$ (as 18T46):
A solvable group of order 108 |
The 27 conjugacy class representatives for $C_3\times S_3^2$ |
Character table for $C_3\times S_3^2$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 3.3.756.1, 6.0.1714608.1, 6.0.8680203.6 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 18 siblings: | data not computed |
Degree 27 sibling: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | 12.0.46311824336645839104.4 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
\(3\) | Deg $18$ | $18$ | $1$ | $37$ | |||
\(7\) | 7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
7.6.5.6 | $x^{6} + 28$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |