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Properties

Label 18.0.97597642979...2624.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 31^{9}\cdot 37^{14}$
Root discriminant $146.58$
Ramified primes $2, 31, 37$
Class number $1228608$ (GRH)
Class group $[2, 2, 6, 6, 8532]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1581654073, -1302399605, 1370146814, -728414092, 442941691, -175507995, 78686738, -24753343, 8987378, -2313995, 715866, -152011, 41454, -7347, 1875, -290, 66, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 66*x^16 - 290*x^15 + 1875*x^14 - 7347*x^13 + 41454*x^12 - 152011*x^11 + 715866*x^10 - 2313995*x^9 + 8987378*x^8 - 24753343*x^7 + 78686738*x^6 - 175507995*x^5 + 442941691*x^4 - 728414092*x^3 + 1370146814*x^2 - 1302399605*x + 1581654073)
 
gp: K = bnfinit(x^18 - 7*x^17 + 66*x^16 - 290*x^15 + 1875*x^14 - 7347*x^13 + 41454*x^12 - 152011*x^11 + 715866*x^10 - 2313995*x^9 + 8987378*x^8 - 24753343*x^7 + 78686738*x^6 - 175507995*x^5 + 442941691*x^4 - 728414092*x^3 + 1370146814*x^2 - 1302399605*x + 1581654073, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 66 x^{16} - 290 x^{15} + 1875 x^{14} - 7347 x^{13} + 41454 x^{12} - 152011 x^{11} + 715866 x^{10} - 2313995 x^{9} + 8987378 x^{8} - 24753343 x^{7} + 78686738 x^{6} - 175507995 x^{5} + 442941691 x^{4} - 728414092 x^{3} + 1370146814 x^{2} - 1302399605 x + 1581654073 \)

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

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:  \(-975976429792931978524482994492749082624=-\,2^{12}\cdot 31^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $146.58$
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:  $2, 31, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{279449506180256538668925569751226646264274409927206696739} a^{17} + \frac{57411544981427054938221481830538720205217144754875206321}{279449506180256538668925569751226646264274409927206696739} a^{16} + \frac{119522064205989298397453474356137258473953845932150678427}{279449506180256538668925569751226646264274409927206696739} a^{15} + \frac{110193008665732349797090552140055894390442493554877539524}{279449506180256538668925569751226646264274409927206696739} a^{14} - \frac{12004954590634070877935344132322156815464948346633689988}{279449506180256538668925569751226646264274409927206696739} a^{13} + \frac{132226253257268353324891643988667727208619743924469689683}{279449506180256538668925569751226646264274409927206696739} a^{12} + \frac{106352121445121643015404689415354036845004986207639992914}{279449506180256538668925569751226646264274409927206696739} a^{11} - \frac{3106744784649257134000346146166943505588895617752256947}{279449506180256538668925569751226646264274409927206696739} a^{10} + \frac{35428009308668713024166933962185668996080277547861871694}{279449506180256538668925569751226646264274409927206696739} a^{9} + \frac{69418214748114913870176563262149102277909284581928345437}{279449506180256538668925569751226646264274409927206696739} a^{8} + \frac{132517552243604798814825515570695196159175317684839568527}{279449506180256538668925569751226646264274409927206696739} a^{7} + \frac{15712768226687115449001034295803599202508146526143932014}{279449506180256538668925569751226646264274409927206696739} a^{6} - \frac{53473670959406038844101043097990686931461524405015039752}{279449506180256538668925569751226646264274409927206696739} a^{5} + \frac{128473878640328044427966545819390358224702932087722923155}{279449506180256538668925569751226646264274409927206696739} a^{4} - \frac{54813717684989522927787825153885676809125599216462338913}{279449506180256538668925569751226646264274409927206696739} a^{3} + \frac{23874270636135696052145622979250369110143407576864078132}{279449506180256538668925569751226646264274409927206696739} a^{2} - \frac{88224752158771881403975617331225353794957959577575714735}{279449506180256538668925569751226646264274409927206696739} a + \frac{89869657206343803784962387645945895972890066711347192939}{279449506180256538668925569751226646264274409927206696739}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6}\times C_{6}\times C_{8532}$, which has order $1228608$ (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:  $8$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 615797.1340659427 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

\(\Q(\sqrt{-31}) \), 3.3.148.1, 3.3.1369.1, 6.0.652542064.2, 6.0.55833130351.2, 9.9.6075640136512.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ R R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$37$37.6.4.1$x^{6} + 518 x^{3} + 171125$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
37.12.10.1$x^{12} + 1998 x^{6} + 21390625$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$