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Properties

Label 18.0.79504772150...5991.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,19^{17}\cdot 29^{9}$
Root discriminant $86.88$
Ramified primes $19, 29$
Class number $376922$ (GRH)
Class group $[376922]$ (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![3787525057, -3020806524, 3020806524, -1377838239, 1377838239, -345115317, 345115317, -50051625, 50051625, -4387006, 4387006, -235677, 235677, -7582, 7582, -134, 134, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 134*x^16 - 134*x^15 + 7582*x^14 - 7582*x^13 + 235677*x^12 - 235677*x^11 + 4387006*x^10 - 4387006*x^9 + 50051625*x^8 - 50051625*x^7 + 345115317*x^6 - 345115317*x^5 + 1377838239*x^4 - 1377838239*x^3 + 3020806524*x^2 - 3020806524*x + 3787525057)
 
gp: K = bnfinit(x^18 - x^17 + 134*x^16 - 134*x^15 + 7582*x^14 - 7582*x^13 + 235677*x^12 - 235677*x^11 + 4387006*x^10 - 4387006*x^9 + 50051625*x^8 - 50051625*x^7 + 345115317*x^6 - 345115317*x^5 + 1377838239*x^4 - 1377838239*x^3 + 3020806524*x^2 - 3020806524*x + 3787525057, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 134 x^{16} - 134 x^{15} + 7582 x^{14} - 7582 x^{13} + 235677 x^{12} - 235677 x^{11} + 4387006 x^{10} - 4387006 x^{9} + 50051625 x^{8} - 50051625 x^{7} + 345115317 x^{6} - 345115317 x^{5} + 1377838239 x^{4} - 1377838239 x^{3} + 3020806524 x^{2} - 3020806524 x + 3787525057 \)

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:  \(-79504772150118146627331158545105991=-\,19^{17}\cdot 29^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.88$
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:  $19, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(551=19\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{551}(320,·)$, $\chi_{551}(1,·)$, $\chi_{551}(260,·)$, $\chi_{551}(521,·)$, $\chi_{551}(202,·)$, $\chi_{551}(465,·)$, $\chi_{551}(86,·)$, $\chi_{551}(349,·)$, $\chi_{551}(30,·)$, $\chi_{551}(291,·)$, $\chi_{551}(550,·)$, $\chi_{551}(231,·)$, $\chi_{551}(233,·)$, $\chi_{551}(173,·)$, $\chi_{551}(175,·)$, $\chi_{551}(376,·)$, $\chi_{551}(378,·)$, $\chi_{551}(318,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{704442593} a^{10} + \frac{205178485}{704442593} a^{9} + \frac{70}{704442593} a^{8} + \frac{246277881}{704442593} a^{7} + \frac{1715}{704442593} a^{6} + \frac{240737350}{704442593} a^{5} + \frac{17150}{704442593} a^{4} + \frac{72159429}{704442593} a^{3} + \frac{60025}{704442593} a^{2} - \frac{59797977}{704442593} a + \frac{33614}{704442593}$, $\frac{1}{704442593} a^{11} + \frac{77}{704442593} a^{9} - \frac{27364209}{704442593} a^{8} + \frac{2156}{704442593} a^{7} - \frac{123510518}{704442593} a^{6} + \frac{26411}{704442593} a^{5} - \frac{48106286}{704442593} a^{4} + \frac{132055}{704442593} a^{3} - \frac{128506683}{704442593} a^{2} + \frac{184877}{704442593} a + \frac{327833273}{704442593}$, $\frac{1}{704442593} a^{12} - \frac{328370508}{704442593} a^{9} - \frac{3234}{704442593} a^{8} - \frac{66957344}{704442593} a^{7} - \frac{105644}{704442593} a^{6} - \frac{269374818}{704442593} a^{5} - \frac{1188495}{704442593} a^{4} - \frac{49241972}{704442593} a^{3} - \frac{4437048}{704442593} a^{2} + \frac{1179351}{704442593} a - \frac{2588278}{704442593}$, $\frac{1}{704442593} a^{13} - \frac{3822}{704442593} a^{9} - \frac{327627353}{704442593} a^{8} - \frac{142688}{704442593} a^{7} + \frac{36414595}{704442593} a^{6} - \frac{1966419}{704442593} a^{5} + \frac{190881786}{704442593} a^{4} - \frac{10487568}{704442593} a^{3} + \frac{137169911}{704442593} a^{2} - \frac{15294370}{704442593} a - \frac{64733805}{704442593}$, $\frac{1}{704442593} a^{14} - \frac{180063692}{704442593} a^{9} + \frac{124852}{704442593} a^{8} + \frac{175171529}{704442593} a^{7} + \frac{4588311}{704442593} a^{6} + \frac{287007028}{704442593} a^{5} + \frac{55059732}{704442593} a^{4} - \frac{210988907}{704442593} a^{3} + \frac{214121180}{704442593} a^{2} + \frac{331240826}{704442593} a + \frac{128472708}{704442593}$, $\frac{1}{704442593} a^{15} + \frac{156065}{704442593} a^{9} + \frac{99663295}{704442593} a^{8} + \frac{6554730}{704442593} a^{7} - \frac{154059519}{704442593} a^{6} + \frac{96354531}{704442593} a^{5} + \frac{309443774}{704442593} a^{4} - \frac{169139643}{704442593} a^{3} - \frac{312793866}{704442593} a^{2} + \frac{98511832}{704442593} a + \frac{90183832}{704442593}$, $\frac{1}{704442593} a^{16} + \frac{61909178}{704442593} a^{9} - \frac{4369820}{704442593} a^{8} + \frac{285201482}{704442593} a^{7} - \frac{171296944}{704442593} a^{6} - \frac{328271507}{704442593} a^{5} - \frac{27884021}{704442593} a^{4} + \frac{49653540}{704442593} a^{3} - \frac{111536084}{704442593} a^{2} + \frac{5992273}{704442593} a - \frac{314870759}{704442593}$, $\frac{1}{704442593} a^{17} - \frac{5714380}{704442593} a^{9} + \frac{178214580}{704442593} a^{8} - \frac{256004224}{704442593} a^{7} - \frac{131680234}{704442593} a^{6} + \frac{306590878}{704442593} a^{5} - \frac{97761509}{704442593} a^{4} + \frac{141793376}{704442593} a^{3} - \frac{157739102}{704442593} a^{2} + \frac{217121107}{704442593} a - \frac{91689570}{704442593}$

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

Class group and class number

$C_{376922}$, which has order $376922$ (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:  \( 22305.8950792 \) (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

\(\Q(\sqrt{-551}) \), 3.3.361.1, 6.0.60389578511.1, \(\Q(\zeta_{19})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ $18$ $18$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ $18$ $18$ $18$

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
19Data not computed
29Data not computed