↑

Properties

Label 18.6.19243278431...5369.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{6}\cdot 1129^{8}$
Root discriminant $32.79$
Ramified primes $3, 1129$
Class number $1$
Class group Trivial
Galois group $C_2^2:D_9$ (as 18T38)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-121, -77, 1087, 1053, 1215, -97, -2306, 1739, -5, -713, 447, -319, -89, 73, -59, -5, 1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + x^16 - 5*x^15 - 59*x^14 + 73*x^13 - 89*x^12 - 319*x^11 + 447*x^10 - 713*x^9 - 5*x^8 + 1739*x^7 - 2306*x^6 - 97*x^5 + 1215*x^4 + 1053*x^3 + 1087*x^2 - 77*x - 121)
 
gp: K = bnfinit(x^18 - 2*x^17 + x^16 - 5*x^15 - 59*x^14 + 73*x^13 - 89*x^12 - 319*x^11 + 447*x^10 - 713*x^9 - 5*x^8 + 1739*x^7 - 2306*x^6 - 97*x^5 + 1215*x^4 + 1053*x^3 + 1087*x^2 - 77*x - 121, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + x^{16} - 5 x^{15} - 59 x^{14} + 73 x^{13} - 89 x^{12} - 319 x^{11} + 447 x^{10} - 713 x^{9} - 5 x^{8} + 1739 x^{7} - 2306 x^{6} - 97 x^{5} + 1215 x^{4} + 1053 x^{3} + 1087 x^{2} - 77 x - 121 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1924327843133559958173465369=3^{6}\cdot 1129^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.79$
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:  $3, 1129$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} + \frac{5}{11} a^{11} - \frac{3}{11} a^{9} - \frac{2}{11} a^{8} - \frac{2}{11} a^{7} + \frac{3}{11} a^{6} - \frac{5}{11} a^{5} - \frac{3}{11} a^{4} + \frac{3}{11} a^{3} - \frac{4}{11} a^{2} - \frac{3}{11} a$, $\frac{1}{11} a^{14} + \frac{5}{11} a^{12} - \frac{3}{11} a^{10} - \frac{2}{11} a^{9} - \frac{2}{11} a^{8} + \frac{3}{11} a^{7} - \frac{5}{11} a^{6} - \frac{3}{11} a^{5} + \frac{3}{11} a^{4} - \frac{4}{11} a^{3} - \frac{3}{11} a^{2}$, $\frac{1}{11} a^{15} + \frac{5}{11} a^{11} - \frac{2}{11} a^{10} + \frac{2}{11} a^{9} + \frac{2}{11} a^{8} + \frac{5}{11} a^{7} + \frac{4}{11} a^{6} - \frac{5}{11} a^{5} + \frac{4}{11} a^{3} - \frac{2}{11} a^{2} + \frac{4}{11} a$, $\frac{1}{979} a^{16} + \frac{26}{979} a^{14} - \frac{42}{979} a^{13} - \frac{294}{979} a^{12} + \frac{393}{979} a^{11} + \frac{111}{979} a^{10} + \frac{472}{979} a^{9} - \frac{106}{979} a^{8} - \frac{428}{979} a^{7} - \frac{129}{979} a^{6} - \frac{29}{89} a^{5} + \frac{417}{979} a^{4} - \frac{474}{979} a^{3} + \frac{457}{979} a^{2} - \frac{182}{979} a + \frac{4}{89}$, $\frac{1}{1259597455542960495429614443} a^{17} - \frac{270079785737139466742991}{1259597455542960495429614443} a^{16} + \frac{22146418542893875578851324}{1259597455542960495429614443} a^{15} + \frac{40816356198181293003361532}{1259597455542960495429614443} a^{14} - \frac{24784854699309096433939208}{1259597455542960495429614443} a^{13} + \frac{478895263151983504769062042}{1259597455542960495429614443} a^{12} - \frac{61620699205594373747396016}{1259597455542960495429614443} a^{11} - \frac{253663865588377319169575590}{1259597455542960495429614443} a^{10} + \frac{474477806707774741077858607}{1259597455542960495429614443} a^{9} - \frac{508420842376804687141281622}{1259597455542960495429614443} a^{8} - \frac{501890005809515174955215316}{1259597455542960495429614443} a^{7} - \frac{259466310205883380573306260}{1259597455542960495429614443} a^{6} - \frac{19976521827172893090048447}{114508859594814590493601313} a^{5} - \frac{383016556659849985235770937}{1259597455542960495429614443} a^{4} - \frac{101412854650460147175638350}{1259597455542960495429614443} a^{3} + \frac{251327172140728008453642718}{1259597455542960495429614443} a^{2} + \frac{299407387340428644856364622}{1259597455542960495429614443} a + \frac{33129836612110172515980273}{114508859594814590493601313}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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:  \( 5000532.57269 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_9$ (as 18T38):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $C_2^2:D_9$
Character table for $C_2^2:D_9$

Intermediate fields

3.3.1129.1, 6.2.11471769.2, 9.9.1624709678881.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
1129Data not computed