Normalized defining polynomial
\( x^{18} - 12x^{15} + 87x^{12} - 848x^{9} + 7392x^{6} - 18816x^{3} + 21952 \)
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: | $2\cdot 3^{121/54}7^{5/6}\approx 118.68160094629725$ | ||
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{3}+\frac{1}{3}$, $\frac{1}{12}a^{10}+\frac{1}{4}a^{4}-\frac{1}{3}a$, $\frac{1}{36}a^{11}+\frac{1}{36}a^{10}-\frac{1}{18}a^{9}-\frac{1}{6}a^{8}-\frac{1}{6}a^{7}-\frac{1}{6}a^{6}+\frac{1}{12}a^{5}-\frac{5}{12}a^{4}+\frac{1}{3}a^{3}-\frac{5}{18}a^{2}+\frac{2}{9}a-\frac{4}{9}$, $\frac{1}{1008}a^{12}-\frac{5}{126}a^{9}-\frac{9}{112}a^{6}-\frac{37}{252}a^{3}-\frac{1}{18}$, $\frac{1}{2016}a^{13}-\frac{5}{252}a^{10}-\frac{9}{224}a^{7}+\frac{215}{504}a^{4}-\frac{1}{36}a$, $\frac{1}{2016}a^{14}+\frac{1}{126}a^{11}+\frac{1}{36}a^{10}-\frac{1}{18}a^{9}-\frac{139}{672}a^{8}-\frac{1}{6}a^{7}-\frac{1}{6}a^{6}+\frac{5}{504}a^{5}-\frac{5}{12}a^{4}+\frac{1}{3}a^{3}+\frac{7}{36}a^{2}+\frac{2}{9}a-\frac{4}{9}$, $\frac{1}{12672576}a^{15}+\frac{55}{132006}a^{12}-\frac{747457}{12672576}a^{9}-\frac{248887}{3168144}a^{6}+\frac{10565}{75432}a^{3}+\frac{1015}{4041}$, $\frac{1}{12672576}a^{16}-\frac{503}{6336288}a^{13}-\frac{55113}{1408064}a^{10}-\frac{243191}{6336288}a^{7}-\frac{8105}{28287}a^{4}+\frac{501}{1796}a$, $\frac{1}{177416064}a^{17}-\frac{2483}{22177008}a^{14}-\frac{1149761}{177416064}a^{11}+\frac{1}{36}a^{10}-\frac{1}{18}a^{9}-\frac{5547985}{44354016}a^{8}-\frac{1}{6}a^{7}-\frac{1}{6}a^{6}-\frac{45247}{452592}a^{5}-\frac{5}{12}a^{4}+\frac{1}{3}a^{3}+\frac{7018}{28287}a^{2}+\frac{2}{9}a-\frac{4}{9}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | \( -\frac{37}{258624} a^{15} + \frac{101}{64656} a^{12} - \frac{2491}{258624} a^{9} + \frac{3731}{32328} a^{6} - \frac{30919}{32328} a^{3} + \frac{15329}{8082} \) (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: | $\frac{943}{12672576}a^{15}-\frac{3077}{1584072}a^{12}+\frac{28411}{4224192}a^{9}-\frac{314359}{3168144}a^{6}+\frac{229241}{226296}a^{3}-\frac{1837}{1347}$, $\frac{1153}{12672576}a^{17}+\frac{51}{1408064}a^{16}+\frac{117}{1408064}a^{15}-\frac{7933}{6336288}a^{14}-\frac{719}{3168144}a^{13}-\frac{461}{264012}a^{12}+\frac{118399}{12672576}a^{11}+\frac{32261}{12672576}a^{10}+\frac{45377}{4224192}a^{9}-\frac{414497}{6336288}a^{8}-\frac{8759}{176008}a^{7}-\frac{15599}{352016}a^{6}+\frac{14789}{28287}a^{5}+\frac{84817}{226296}a^{4}+\frac{13093}{75432}a^{3}-\frac{14911}{16164}a^{2}-\frac{5293}{8082}a-\frac{1139}{1347}$, $\frac{299}{25345152}a^{17}-\frac{269}{6336288}a^{16}+\frac{117}{1408064}a^{15}+\frac{1805}{6336288}a^{14}+\frac{1153}{2112096}a^{13}-\frac{461}{264012}a^{12}-\frac{110347}{25345152}a^{11}-\frac{35779}{6336288}a^{10}+\frac{45377}{4224192}a^{9}+\frac{87803}{3168144}a^{8}+\frac{405401}{6336288}a^{7}-\frac{15599}{352016}a^{6}-\frac{60853}{452592}a^{5}-\frac{27019}{75432}a^{4}+\frac{13093}{75432}a^{3}+\frac{11147}{16164}a^{2}+\frac{4595}{16164}a-\frac{1139}{1347}$, $\frac{71}{8448384}a^{17}-\frac{1019}{12672576}a^{16}-\frac{55}{528024}a^{15}-\frac{95}{352016}a^{14}+\frac{363}{352016}a^{13}+\frac{655}{1056048}a^{12}+\frac{34681}{8448384}a^{11}-\frac{53653}{12672576}a^{10}-\frac{305}{528024}a^{9}-\frac{1031}{2112096}a^{8}+\frac{124597}{1584072}a^{7}+\frac{105593}{1056048}a^{6}+\frac{1965}{7184}a^{5}-\frac{11545}{25144}a^{4}+\frac{5009}{37716}a^{3}-\frac{1010}{1347}a^{2}+\frac{4463}{8082}a-\frac{139}{2694}$, $\frac{1753}{5544252}a^{17}-\frac{835}{2112096}a^{16}-\frac{829}{3168144}a^{15}-\frac{8005}{2772126}a^{14}+\frac{27631}{6336288}a^{13}+\frac{2111}{1584072}a^{12}+\frac{112583}{5544252}a^{11}-\frac{194471}{6336288}a^{10}-\frac{32027}{3168144}a^{9}-\frac{613331}{2772126}a^{8}+\frac{222085}{704032}a^{7}+\frac{239615}{1584072}a^{6}+\frac{50945}{28287}a^{5}-\frac{648313}{226296}a^{4}-\frac{53555}{56574}a^{3}-\frac{23024}{28287}a^{2}+\frac{90247}{16164}a-\frac{10363}{4041}$, $\frac{7897}{22177008}a^{17}-\frac{6157}{12672576}a^{16}+\frac{829}{3168144}a^{15}-\frac{43441}{11088504}a^{14}+\frac{3267}{704032}a^{13}-\frac{2111}{1584072}a^{12}+\frac{65567}{2464112}a^{11}-\frac{138625}{4224192}a^{10}+\frac{32027}{3168144}a^{9}-\frac{3054461}{11088504}a^{8}+\frac{2264747}{6336288}a^{7}-\frac{239615}{1584072}a^{6}+\frac{70487}{28287}a^{5}-\frac{27031}{9429}a^{4}+\frac{53555}{56574}a^{3}-\frac{39187}{9429}a^{2}+\frac{4463}{1796}a+\frac{10363}{4041}$, $\frac{92011}{88708032}a^{17}+\frac{2263}{1584072}a^{16}+\frac{36073}{12672576}a^{15}-\frac{437735}{44354016}a^{14}-\frac{1455}{88004}a^{13}-\frac{5113}{198009}a^{12}+\frac{5989349}{88708032}a^{11}+\frac{49709}{528024}a^{10}+\frac{2113271}{12672576}a^{9}-\frac{29498243}{44354016}a^{8}-\frac{856829}{792036}a^{7}-\frac{6444967}{3168144}a^{6}+\frac{684787}{113148}a^{5}+\frac{51389}{5388}a^{4}+\frac{3267059}{226296}a^{3}-\frac{431237}{113148}a^{2}-\frac{12344}{1347}a-\frac{70103}{4041}$, $\frac{70937}{88708032}a^{17}+\frac{6203}{6336288}a^{16}+\frac{223}{176008}a^{15}-\frac{84569}{11088504}a^{14}-\frac{57937}{6336288}a^{13}-\frac{3349}{264012}a^{12}+\frac{4327271}{88708032}a^{11}+\frac{354413}{6336288}a^{10}+\frac{40283}{528024}a^{9}-\frac{12412349}{22177008}a^{8}-\frac{4320611}{6336288}a^{7}-\frac{83989}{88004}a^{6}+\frac{972473}{226296}a^{5}+\frac{1117573}{226296}a^{4}+\frac{117947}{18858}a^{3}-\frac{134387}{28287}a^{2}-\frac{90709}{16164}a-\frac{13192}{1347}$ (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)]
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Regulator: | \( 174512587.28025168 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 174512587.28025168 \cdot 3}{6\cdot\sqrt{520986863358984384396363313152}}\cr\approx \mathstrut & 1.84502472510635 \end{aligned}\] (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 $C_3\times S_3^2$ is not computed |
Intermediate fields
\(\Q(\sqrt{-3}) \), 3.1.756.1, 6.0.8680203.5, 6.0.1714608.3 |
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: | 12.0.740989189386333425664.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
\(3\) | Deg $18$ | $18$ | $1$ | $37$ | |||
\(7\) | 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
7.6.5.2 | $x^{6} + 42$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
7.6.3.1 | $x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |