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Properties

Label 18.0.36928908831...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{45}\cdot 5^{15}$
Root discriminant $94.62$
Ramified primes $2, 3, 5$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22056, 28080, 49680, -1944, -32400, -32940, -9180, 6480, 11340, 5805, 3420, 1080, 315, 0, -90, -9, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^15 - 90*x^14 + 315*x^12 + 1080*x^11 + 3420*x^10 + 5805*x^9 + 11340*x^8 + 6480*x^7 - 9180*x^6 - 32940*x^5 - 32400*x^4 - 1944*x^3 + 49680*x^2 + 28080*x + 22056)
 
gp: K = bnfinit(x^18 - 9*x^15 - 90*x^14 + 315*x^12 + 1080*x^11 + 3420*x^10 + 5805*x^9 + 11340*x^8 + 6480*x^7 - 9180*x^6 - 32940*x^5 - 32400*x^4 - 1944*x^3 + 49680*x^2 + 28080*x + 22056, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{15} - 90 x^{14} + 315 x^{12} + 1080 x^{11} + 3420 x^{10} + 5805 x^{9} + 11340 x^{8} + 6480 x^{7} - 9180 x^{6} - 32940 x^{5} - 32400 x^{4} - 1944 x^{3} + 49680 x^{2} + 28080 x + 22056 \)

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:  \(-369289088318854212330375000000000000=-\,2^{12}\cdot 3^{45}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $94.62$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{6}$, $\frac{1}{292} a^{16} - \frac{25}{292} a^{15} - \frac{16}{73} a^{14} + \frac{55}{292} a^{13} - \frac{1}{292} a^{12} + \frac{45}{146} a^{11} - \frac{5}{292} a^{10} + \frac{77}{292} a^{9} - \frac{6}{73} a^{8} + \frac{145}{292} a^{7} + \frac{15}{292} a^{6} - \frac{17}{73} a^{5} - \frac{19}{73} a^{4} + \frac{11}{73} a^{3} - \frac{29}{73} a^{2} + \frac{33}{73} a - \frac{33}{73}$, $\frac{1}{130344573018927980677505847370739981428} a^{17} + \frac{107915088981889184684775325345877361}{65172286509463990338752923685369990714} a^{16} + \frac{11591043295872192508743991003625630873}{130344573018927980677505847370739981428} a^{15} - \frac{2493235292677512665555658585161308171}{130344573018927980677505847370739981428} a^{14} + \frac{7074643531951705778425427592817283221}{65172286509463990338752923685369990714} a^{13} + \frac{13789830230377868789751783752638119541}{130344573018927980677505847370739981428} a^{12} + \frac{60932822835996274408739564723098929583}{130344573018927980677505847370739981428} a^{11} - \frac{13588376221631824939987715252904145952}{32586143254731995169376461842684995357} a^{10} - \frac{16624037200712662193546412874563362135}{130344573018927980677505847370739981428} a^{9} + \frac{52320342146555857788282192709391844047}{130344573018927980677505847370739981428} a^{8} + \frac{14723979091454967996454271851374676337}{32586143254731995169376461842684995357} a^{7} - \frac{62195315225028720769561288958869569029}{130344573018927980677505847370739981428} a^{6} - \frac{29010822210031814031046186949856137523}{65172286509463990338752923685369990714} a^{5} + \frac{20731171237329171620584497704314582955}{65172286509463990338752923685369990714} a^{4} - \frac{10773936348904751566665141219775469725}{65172286509463990338752923685369990714} a^{3} - \frac{6705417493096087589099954929890298358}{32586143254731995169376461842684995357} a^{2} - \frac{6808423768143553170204303323968960415}{32586143254731995169376461842684995357} a + \frac{2262350075450311957946780408096322596}{32586143254731995169376461842684995357}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 33562079435.249725 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.1.243.1, 6.0.22143375.1, 9.1.31381059609000000.8

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

Sibling fields

Degree 18 siblings: 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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$5$5.6.5.2$x^{6} + 10$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.12.10.2$x^{12} + 15 x^{6} + 100$$6$$2$$10$$C_6\times S_3$$[\ ]_{6}^{6}$