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Properties

Label 18.0.75855608471...6739.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 19^{17}$
Root discriminant $59.04$
Ramified primes $7, 19$
Class number $243$ (GRH)
Class group $[3, 3, 3, 3, 3]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32263, -77749, 418761, -597570, 245519, 3723, 23181, -2167, -14857, -1673, 951, 1177, 476, -39, 39, -39, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + x^16 - 39*x^15 + 39*x^14 - 39*x^13 + 476*x^12 + 1177*x^11 + 951*x^10 - 1673*x^9 - 14857*x^8 - 2167*x^7 + 23181*x^6 + 3723*x^5 + 245519*x^4 - 597570*x^3 + 418761*x^2 - 77749*x + 32263)
 
gp: K = bnfinit(x^18 - x^17 + x^16 - 39*x^15 + 39*x^14 - 39*x^13 + 476*x^12 + 1177*x^11 + 951*x^10 - 1673*x^9 - 14857*x^8 - 2167*x^7 + 23181*x^6 + 3723*x^5 + 245519*x^4 - 597570*x^3 + 418761*x^2 - 77749*x + 32263, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + x^{16} - 39 x^{15} + 39 x^{14} - 39 x^{13} + 476 x^{12} + 1177 x^{11} + 951 x^{10} - 1673 x^{9} - 14857 x^{8} - 2167 x^{7} + 23181 x^{6} + 3723 x^{5} + 245519 x^{4} - 597570 x^{3} + 418761 x^{2} - 77749 x + 32263 \)

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:  \(-75855608471185389708554302866739=-\,7^{12}\cdot 19^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.04$
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:  $7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(133=7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{133}(64,·)$, $\chi_{133}(1,·)$, $\chi_{133}(2,·)$, $\chi_{133}(67,·)$, $\chi_{133}(4,·)$, $\chi_{133}(8,·)$, $\chi_{133}(128,·)$, $\chi_{133}(79,·)$, $\chi_{133}(16,·)$, $\chi_{133}(25,·)$, $\chi_{133}(93,·)$, $\chi_{133}(32,·)$, $\chi_{133}(100,·)$, $\chi_{133}(106,·)$, $\chi_{133}(113,·)$, $\chi_{133}(50,·)$, $\chi_{133}(53,·)$, $\chi_{133}(123,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{11} a^{9} + \frac{1}{11} a^{7} + \frac{1}{11} a^{5} + \frac{1}{11} a^{3} + \frac{1}{11} a$, $\frac{1}{11} a^{10} + \frac{1}{11} a^{8} + \frac{1}{11} a^{6} + \frac{1}{11} a^{4} + \frac{1}{11} a^{2}$, $\frac{1}{11} a^{11} - \frac{1}{11} a$, $\frac{1}{11} a^{12} - \frac{1}{11} a^{2}$, $\frac{1}{11} a^{13} - \frac{1}{11} a^{3}$, $\frac{1}{847} a^{14} - \frac{38}{847} a^{13} - \frac{5}{121} a^{12} - \frac{23}{847} a^{11} + \frac{10}{847} a^{10} - \frac{6}{847} a^{9} - \frac{342}{847} a^{8} - \frac{270}{847} a^{7} + \frac{131}{847} a^{6} + \frac{29}{121} a^{5} - \frac{178}{847} a^{4} - \frac{419}{847} a^{3} + \frac{19}{121} a^{2} - \frac{29}{121} a + \frac{5}{11}$, $\frac{1}{847} a^{15} - \frac{16}{847} a^{13} + \frac{3}{77} a^{12} - \frac{17}{847} a^{11} - \frac{1}{77} a^{10} - \frac{31}{847} a^{9} - \frac{9}{77} a^{8} - \frac{39}{121} a^{7} - \frac{26}{77} a^{6} - \frac{395}{847} a^{5} + \frac{5}{77} a^{4} + \frac{227}{847} a^{3} - \frac{4}{11} a^{2} - \frac{2}{121} a + \frac{3}{11}$, $\frac{1}{847} a^{16} - \frac{36}{847} a^{13} - \frac{38}{847} a^{12} + \frac{6}{847} a^{11} - \frac{25}{847} a^{10} + \frac{36}{847} a^{9} + \frac{30}{847} a^{8} - \frac{20}{121} a^{7} - \frac{21}{121} a^{6} + \frac{146}{847} a^{5} - \frac{234}{847} a^{4} + \frac{303}{847} a^{3} - \frac{39}{121} a^{2} + \frac{31}{121} a + \frac{3}{11}$, $\frac{1}{667491879014678243581845437271403241} a^{17} - \frac{388832377335511072610494008030704}{667491879014678243581845437271403241} a^{16} - \frac{295286835163965095629083896062685}{667491879014678243581845437271403241} a^{15} - \frac{1212510717222343847586583954069}{8668725701489327838725265419109133} a^{14} - \frac{9366122995974516223557427092159196}{667491879014678243581845437271403241} a^{13} + \frac{9321316838773802774595475609257215}{667491879014678243581845437271403241} a^{12} - \frac{9600921545697916424410761806593652}{667491879014678243581845437271403241} a^{11} + \frac{75591724898253966150605708164553}{95355982716382606225977919610200463} a^{10} - \frac{9896026543305249271367127721634291}{667491879014678243581845437271403241} a^{9} + \frac{312036951440422798104074620296061295}{667491879014678243581845437271403241} a^{8} - \frac{1015495725567477661727896852062252}{6238241859950263958708835862349563} a^{7} + \frac{20919604128945627859411964185960147}{95355982716382606225977919610200463} a^{6} - \frac{15326334588958248355142516799221817}{667491879014678243581845437271403241} a^{5} + \frac{153087272047008791584003085879445078}{667491879014678243581845437271403241} a^{4} + \frac{1620432824622019680672760185207938}{6238241859950263958708835862349563} a^{3} + \frac{14393408382653946801347847779347316}{95355982716382606225977919610200463} a^{2} + \frac{41404533970979703643824554809584904}{95355982716382606225977919610200463} a - \frac{3288411220912217401888055279562117}{8668725701489327838725265419109133}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}$, which has order $243$ (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:  \( 7595459.56275 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-19}) \), 3.3.361.1, 6.0.2476099.1, 9.9.1998099208210609.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $18$ $18$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.1.0.1}{1} }^{18}$ $18$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ $18$

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
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
19Data not computed