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Properties

Label 18.12.3007309704...0000.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{12}\cdot 3^{24}\cdot 5^{9}\cdot 11^{3}$
Root discriminant $22.90$
Ramified primes $2, 3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T189

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16, 0, 240, 32, -1104, -72, 2352, -276, -2604, 902, 1350, -840, -139, 246, -93, 14, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 9*x^16 + 14*x^15 - 93*x^14 + 246*x^13 - 139*x^12 - 840*x^11 + 1350*x^10 + 902*x^9 - 2604*x^8 - 276*x^7 + 2352*x^6 - 72*x^5 - 1104*x^4 + 32*x^3 + 240*x^2 - 16)
 
gp: K = bnfinit(x^18 - 6*x^17 + 9*x^16 + 14*x^15 - 93*x^14 + 246*x^13 - 139*x^12 - 840*x^11 + 1350*x^10 + 902*x^9 - 2604*x^8 - 276*x^7 + 2352*x^6 - 72*x^5 - 1104*x^4 + 32*x^3 + 240*x^2 - 16, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 9 x^{16} + 14 x^{15} - 93 x^{14} + 246 x^{13} - 139 x^{12} - 840 x^{11} + 1350 x^{10} + 902 x^{9} - 2604 x^{8} - 276 x^{7} + 2352 x^{6} - 72 x^{5} - 1104 x^{4} + 32 x^{3} + 240 x^{2} - 16 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-3007309704449688000000000=-\,2^{12}\cdot 3^{24}\cdot 5^{9}\cdot 11^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.90$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{6}$, $\frac{1}{56} a^{15} + \frac{1}{14} a^{14} - \frac{1}{56} a^{13} + \frac{1}{14} a^{12} + \frac{13}{56} a^{11} + \frac{1}{14} a^{10} - \frac{5}{56} a^{9} - \frac{1}{28} a^{8} + \frac{1}{28} a^{6} + \frac{3}{14} a^{5} - \frac{1}{14} a^{4} - \frac{2}{7} a^{3} + \frac{2}{7} a^{2} + \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{56} a^{16} - \frac{3}{56} a^{14} - \frac{3}{28} a^{13} - \frac{3}{56} a^{12} - \frac{3}{28} a^{11} + \frac{1}{8} a^{10} + \frac{1}{14} a^{9} + \frac{1}{7} a^{8} - \frac{3}{14} a^{7} - \frac{5}{28} a^{6} + \frac{1}{14} a^{5} - \frac{1}{2} a^{4} + \frac{3}{7} a^{3} - \frac{2}{7}$, $\frac{1}{27944} a^{17} - \frac{31}{13972} a^{16} - \frac{3}{6986} a^{15} + \frac{421}{6986} a^{14} - \frac{55}{499} a^{13} - \frac{1461}{13972} a^{12} - \frac{459}{3493} a^{11} + \frac{150}{3493} a^{10} - \frac{2477}{27944} a^{9} - \frac{91}{499} a^{8} + \frac{167}{3493} a^{7} - \frac{654}{3493} a^{6} - \frac{2511}{6986} a^{5} + \frac{379}{6986} a^{4} + \frac{279}{998} a^{3} - \frac{788}{3493} a^{2} + \frac{745}{3493} a + \frac{1194}{3493}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $14$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 826735.916468 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T189:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 54 conjugacy class representatives for t18n189 are not computed
Character table for t18n189 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 6.6.13122000.2, 6.4.9021375.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: 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 R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ R ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.12.8.2$x^{12} - 8 x^{3} + 16$$3$$4$$8$$C_3\times (C_3 : C_4)$$[\ ]_{3}^{12}$
3Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$