Normalized defining polynomial
\( x^{18} - 4 x^{17} + 6 x^{16} - 18 x^{15} + 54 x^{14} - 74 x^{13} + 135 x^{12} - 274 x^{11} + 151 x^{10} + 98 x^{9} - 31 x^{8} + 20 x^{7} + 29 x^{6} - 16 x^{5} + 6 x^{4} + 18 x^{3} + 13 x^{2} - 2 x + 1 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-86675003008018355847168=-\,2^{33}\cdot 3^{6}\cdot 7^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $18.81$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 3, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{129} a^{15} + \frac{32}{129} a^{14} + \frac{34}{129} a^{13} - \frac{62}{129} a^{12} - \frac{49}{129} a^{11} - \frac{38}{129} a^{10} - \frac{17}{129} a^{9} - \frac{50}{129} a^{8} - \frac{47}{129} a^{7} + \frac{56}{129} a^{6} + \frac{38}{129} a^{5} + \frac{8}{43} a^{4} + \frac{50}{129} a^{3} + \frac{18}{43} a^{2} + \frac{35}{129} a + \frac{1}{129}$, $\frac{1}{129} a^{16} + \frac{14}{43} a^{14} + \frac{11}{129} a^{13} - \frac{6}{43} a^{11} + \frac{38}{129} a^{10} - \frac{22}{129} a^{9} + \frac{5}{129} a^{8} + \frac{4}{43} a^{7} + \frac{52}{129} a^{6} - \frac{31}{129} a^{5} + \frac{56}{129} a^{4} + \frac{2}{129} a^{3} - \frac{16}{129} a^{2} + \frac{14}{43} a - \frac{32}{129}$, $\frac{1}{506072874077823} a^{17} + \frac{88931764786}{38928682621371} a^{16} + \frac{577510445401}{506072874077823} a^{15} - \frac{42289167745676}{506072874077823} a^{14} + \frac{5929823852710}{12976227540457} a^{13} + \frac{193961441769151}{506072874077823} a^{12} + \frac{117295909560799}{506072874077823} a^{11} - \frac{146271720201784}{506072874077823} a^{10} + \frac{214015197025814}{506072874077823} a^{9} - \frac{68211677146751}{168690958025941} a^{8} - \frac{155381349836320}{506072874077823} a^{7} - \frac{17387217403459}{506072874077823} a^{6} - \frac{67808726474584}{168690958025941} a^{5} + \frac{6014815856401}{506072874077823} a^{4} + \frac{59830138081258}{168690958025941} a^{3} - \frac{34987260465073}{506072874077823} a^{2} + \frac{66544043009192}{168690958025941} a + \frac{204074460781298}{506072874077823}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $8$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 9688.268698021906 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_6\times S_3$ (as 18T6):
| A solvable group of order 36 |
| The 18 conjugacy class representatives for $S_3 \times C_6$ |
| Character table for $S_3 \times C_6$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{7})^+\), 3.1.1176.1, 6.0.44255232.1, 6.0.1229312.1, 9.3.1626379776.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 12 sibling: | data not computed |
| Degree 18 sibling: | data not computed |
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 | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ | R | ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.6.9.5 | $x^{6} - 4 x^{4} + 4 x^{2} + 8$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ |
| 2.12.24.307 | $x^{12} + 28 x^{11} - 2 x^{10} + 16 x^{9} + 26 x^{8} + 8 x^{7} + 20 x^{6} - 24 x^{5} - 8 x^{4} + 32 x^{3} + 32 x^{2} + 32 x + 24$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $[2, 3]^{3}$ | |
| $3$ | 3.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 3.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 3.6.3.1 | $x^{6} - 6 x^{4} + 9 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.1 | $x^{6} - 6 x^{4} + 9 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 7 | Data not computed | ||||||