Normalized defining polynomial
\( x^{18} + 16x^{16} + 102x^{14} + 475x^{12} + 1129x^{10} + 1067x^{8} + 477x^{6} + 136x^{4} + 29x^{2} + 1 \)
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: | \(-1725681370618930208702464\) \(\medspace = -\,2^{18}\cdot 37^{12}\) | 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: | \(22.21\) | 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: | $2\cdot 37^{2/3}\approx 22.20740493717357$ | ||
Ramified primes: | \(2\), \(37\) | 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: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{20}a^{12}-\frac{1}{2}a^{11}+\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{20}a^{8}+\frac{1}{4}a^{6}+\frac{3}{20}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{9}{20}$, $\frac{1}{20}a^{13}+\frac{1}{4}a^{11}+\frac{1}{20}a^{9}-\frac{1}{2}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}+\frac{3}{20}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}+\frac{9}{20}a-\frac{1}{2}$, $\frac{1}{20}a^{14}-\frac{1}{2}a^{11}-\frac{1}{5}a^{10}-\frac{1}{2}a^{7}-\frac{1}{10}a^{6}-\frac{1}{2}a^{5}-\frac{3}{10}a^{2}-\frac{1}{4}$, $\frac{1}{20}a^{15}-\frac{1}{5}a^{11}-\frac{1}{2}a^{10}-\frac{1}{10}a^{7}-\frac{1}{2}a^{4}-\frac{3}{10}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{89484340}a^{16}+\frac{138783}{17896868}a^{14}+\frac{639581}{89484340}a^{12}-\frac{1}{2}a^{11}-\frac{441581}{1626988}a^{10}-\frac{1}{2}a^{9}+\frac{12769643}{89484340}a^{8}-\frac{1}{2}a^{7}+\frac{5420127}{17896868}a^{6}+\frac{18691729}{89484340}a^{4}-\frac{1}{2}a^{3}-\frac{292375}{4474217}a^{2}-\frac{2780903}{8948434}$, $\frac{1}{89484340}a^{17}+\frac{138783}{17896868}a^{15}+\frac{639581}{89484340}a^{13}-\frac{441581}{1626988}a^{11}+\frac{12769643}{89484340}a^{9}+\frac{5420127}{17896868}a^{7}-\frac{1}{2}a^{6}+\frac{18691729}{89484340}a^{5}-\frac{292375}{4474217}a^{3}-\frac{1}{2}a^{2}-\frac{2780903}{8948434}a-\frac{1}{2}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{2}$, which has order $4$
Unit group
Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{15259}{62315} a^{17} + \frac{245612}{62315} a^{15} + \frac{631611}{24926} a^{13} + \frac{1342489}{11330} a^{11} + \frac{35669981}{124630} a^{9} + \frac{35044497}{124630} a^{7} + \frac{15724383}{124630} a^{5} + \frac{3779241}{124630} a^{3} + \frac{744023}{124630} a \) (order $4$) | 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: | $\frac{16699649}{89484340}a^{17}+\frac{340973}{8134940}a^{16}+\frac{64855136}{22371085}a^{15}+\frac{2605059}{4067470}a^{14}+\frac{1582844959}{89484340}a^{13}+\frac{30920553}{8134940}a^{12}+\frac{654426109}{8134940}a^{11}+\frac{12548413}{739540}a^{10}+\frac{15516810247}{89484340}a^{9}+\frac{275459699}{8134940}a^{8}+\frac{10646571407}{89484340}a^{7}+\frac{119487759}{8134940}a^{6}+\frac{2996247561}{89484340}a^{5}-\frac{21112613}{8134940}a^{4}+\frac{274102273}{44742170}a^{3}-\frac{2028859}{4067470}a^{2}+\frac{13480519}{17896868}a-\frac{736245}{1626988}$, $\frac{1597175}{8948434}a^{17}+\frac{2300743}{44742170}a^{16}+\frac{120160219}{44742170}a^{15}+\frac{73360461}{89484340}a^{14}+\frac{139578481}{8948434}a^{13}+\frac{116180744}{22371085}a^{12}+\frac{140729482}{2033735}a^{11}+\frac{48923709}{2033735}a^{10}+\frac{1175494673}{8948434}a^{9}+\frac{1258599937}{22371085}a^{8}+\frac{1042561001}{22371085}a^{7}+\frac{2228028819}{44742170}a^{6}+\frac{20078590}{4474217}a^{5}+\frac{408051861}{22371085}a^{4}+\frac{106764458}{22371085}a^{3}+\frac{111851561}{22371085}a^{2}-\frac{6593813}{4474217}a+\frac{33806407}{17896868}$, $\frac{16699649}{89484340}a^{17}-\frac{340973}{8134940}a^{16}+\frac{64855136}{22371085}a^{15}-\frac{2605059}{4067470}a^{14}+\frac{1582844959}{89484340}a^{13}-\frac{30920553}{8134940}a^{12}+\frac{654426109}{8134940}a^{11}-\frac{12548413}{739540}a^{10}+\frac{15516810247}{89484340}a^{9}-\frac{275459699}{8134940}a^{8}+\frac{10646571407}{89484340}a^{7}-\frac{119487759}{8134940}a^{6}+\frac{2996247561}{89484340}a^{5}+\frac{21112613}{8134940}a^{4}+\frac{274102273}{44742170}a^{3}+\frac{2028859}{4067470}a^{2}+\frac{13480519}{17896868}a+\frac{736245}{1626988}$, $\frac{1597175}{8948434}a^{17}-\frac{2300743}{44742170}a^{16}+\frac{120160219}{44742170}a^{15}-\frac{73360461}{89484340}a^{14}+\frac{139578481}{8948434}a^{13}-\frac{116180744}{22371085}a^{12}+\frac{140729482}{2033735}a^{11}-\frac{48923709}{2033735}a^{10}+\frac{1175494673}{8948434}a^{9}-\frac{1258599937}{22371085}a^{8}+\frac{1042561001}{22371085}a^{7}-\frac{2228028819}{44742170}a^{6}+\frac{20078590}{4474217}a^{5}-\frac{408051861}{22371085}a^{4}+\frac{106764458}{22371085}a^{3}-\frac{111851561}{22371085}a^{2}-\frac{6593813}{4474217}a-\frac{33806407}{17896868}$, $\frac{31843}{124630}a^{17}+\frac{508163}{124630}a^{15}+\frac{3231047}{124630}a^{13}+\frac{1368403}{11330}a^{11}+\frac{35665693}{124630}a^{9}+\frac{33980769}{124630}a^{7}+\frac{16252839}{124630}a^{5}+\frac{1901096}{62315}a^{3}+\frac{450248}{62315}a$, $\frac{1589197}{89484340}a^{17}+\frac{328517}{22371085}a^{16}+\frac{25269133}{89484340}a^{15}+\frac{19494271}{89484340}a^{14}+\frac{155505511}{89484340}a^{13}+\frac{21992557}{17896868}a^{12}+\frac{61497313}{8134940}a^{11}+\frac{43140811}{8134940}a^{10}+\frac{269444831}{17896868}a^{9}+\frac{796149861}{89484340}a^{8}-\frac{153373941}{89484340}a^{7}-\frac{144257347}{89484340}a^{6}-\frac{563165767}{17896868}a^{5}-\frac{608573397}{89484340}a^{4}-\frac{821794999}{44742170}a^{3}-\frac{229487701}{89484340}a^{2}-\frac{34137131}{22371085}a+\frac{12460412}{22371085}$, $\frac{2895179}{89484340}a^{17}-\frac{3396699}{44742170}a^{16}+\frac{48909049}{89484340}a^{15}-\frac{56561589}{44742170}a^{14}+\frac{342072871}{89484340}a^{13}-\frac{763351777}{89484340}a^{12}+\frac{156475499}{8134940}a^{11}-\frac{333681783}{8134940}a^{10}+\frac{4985148439}{89484340}a^{9}-\frac{9711697253}{89484340}a^{8}+\frac{8192423977}{89484340}a^{7}-\frac{11938033279}{89484340}a^{6}+\frac{8515012507}{89484340}a^{5}-\frac{7002058059}{89484340}a^{4}+\frac{921933444}{22371085}a^{3}-\frac{1600764007}{89484340}a^{2}+\frac{101018897}{22371085}a-\frac{66755101}{89484340}$, $\frac{1643709}{22371085}a^{16}+\frac{5386619}{4474217}a^{14}+\frac{354760191}{44742170}a^{12}+\frac{30510805}{813494}a^{10}+\frac{4238249887}{44742170}a^{8}+\frac{920062997}{8948434}a^{6}+\frac{2240780841}{44742170}a^{4}+\frac{135378789}{8948434}a^{2}+\frac{163932557}{44742170}$ | 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: | \( 18639.792218 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 18639.792218 \cdot 4}{4\cdot\sqrt{1725681370618930208702464}}\cr\approx \mathstrut & 0.21656066517 \end{aligned}\]
Galois group
$C_3\times S_3$ (as 18T3):
A solvable group of order 18 |
The 9 conjugacy class representatives for $S_3 \times C_3$ |
Character table for $S_3 \times C_3$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), 3.1.5476.1 x3, 3.3.1369.1, 6.0.119946304.2, 6.0.87616.1 x2, 6.0.119946304.1, 9.3.164206490176.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 sibling: | 6.0.87616.1 |
Degree 9 sibling: | 9.3.164206490176.1 |
Minimal sibling: | 6.0.87616.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{3}$ | ${\href{/padicField/5.3.0.1}{3} }^{6}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.2.0.1}{2} }^{9}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}$ | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.2.0.1}{2} }^{9}$ | R | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{9}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.3.0.1}{3} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
\(37\) | 37.9.6.1 | $x^{9} + 18 x^{7} + 216 x^{6} + 108 x^{5} + 594 x^{4} - 19197 x^{3} + 7776 x^{2} - 50544 x + 381240$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
37.9.6.1 | $x^{9} + 18 x^{7} + 216 x^{6} + 108 x^{5} + 594 x^{4} - 19197 x^{3} + 7776 x^{2} - 50544 x + 381240$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |