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Properties

Label 18.0.64990743984...5683.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,17^{9}\cdot 19^{17}$
Root discriminant $66.52$
Ramified primes $17, 19$
Class number $38836$ (GRH)
Class group $[38836]$ (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![57746701, -52765965, 52765965, -34088205, 34088205, -13542669, 13542669, -3269901, 3269901, -487693, 487693, -45069, 45069, -2509, 2509, -77, 77, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 77*x^16 - 77*x^15 + 2509*x^14 - 2509*x^13 + 45069*x^12 - 45069*x^11 + 487693*x^10 - 487693*x^9 + 3269901*x^8 - 3269901*x^7 + 13542669*x^6 - 13542669*x^5 + 34088205*x^4 - 34088205*x^3 + 52765965*x^2 - 52765965*x + 57746701)
 
gp: K = bnfinit(x^18 - x^17 + 77*x^16 - 77*x^15 + 2509*x^14 - 2509*x^13 + 45069*x^12 - 45069*x^11 + 487693*x^10 - 487693*x^9 + 3269901*x^8 - 3269901*x^7 + 13542669*x^6 - 13542669*x^5 + 34088205*x^4 - 34088205*x^3 + 52765965*x^2 - 52765965*x + 57746701, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 77 x^{16} - 77 x^{15} + 2509 x^{14} - 2509 x^{13} + 45069 x^{12} - 45069 x^{11} + 487693 x^{10} - 487693 x^{9} + 3269901 x^{8} - 3269901 x^{7} + 13542669 x^{6} - 13542669 x^{5} + 34088205 x^{4} - 34088205 x^{3} + 52765965 x^{2} - 52765965 x + 57746701 \)

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:  \(-649907439846766024629709761975683=-\,17^{9}\cdot 19^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.52$
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:  $17, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(323=17\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{323}(256,·)$, $\chi_{323}(1,·)$, $\chi_{323}(322,·)$, $\chi_{323}(67,·)$, $\chi_{323}(135,·)$, $\chi_{323}(137,·)$, $\chi_{323}(203,·)$, $\chi_{323}(273,·)$, $\chi_{323}(84,·)$, $\chi_{323}(288,·)$, $\chi_{323}(33,·)$, $\chi_{323}(290,·)$, $\chi_{323}(35,·)$, $\chi_{323}(239,·)$, $\chi_{323}(50,·)$, $\chi_{323}(120,·)$, $\chi_{323}(186,·)$, $\chi_{323}(188,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{14007941} a^{10} + \frac{5519006}{14007941} a^{9} + \frac{40}{14007941} a^{8} + \frac{2573042}{14007941} a^{7} + \frac{560}{14007941} a^{6} + \frac{2860622}{14007941} a^{5} + \frac{3200}{14007941} a^{4} + \frac{6488124}{14007941} a^{3} + \frac{6400}{14007941} a^{2} - \frac{3420604}{14007941} a + \frac{2048}{14007941}$, $\frac{1}{14007941} a^{11} + \frac{44}{14007941} a^{9} + \frac{5939858}{14007941} a^{8} + \frac{704}{14007941} a^{7} - \frac{6035718}{14007941} a^{6} + \frac{4928}{14007941} a^{5} - \frac{4325416}{14007941} a^{4} + \frac{14080}{14007941} a^{3} + \frac{2968198}{14007941} a^{2} + \frac{11264}{14007941} a + \frac{1484099}{14007941}$, $\frac{1}{14007941} a^{12} + \frac{1238591}{14007941} a^{9} - \frac{1056}{14007941} a^{8} + \frac{6821903}{14007941} a^{7} - \frac{19712}{14007941} a^{6} - \frac{4121315}{14007941} a^{5} - \frac{126720}{14007941} a^{4} - \frac{2350438}{14007941} a^{3} - \frac{270336}{14007941} a^{2} - \frac{2096676}{14007941} a - \frac{90112}{14007941}$, $\frac{1}{14007941} a^{13} - \frac{1248}{14007941} a^{9} - \frac{697914}{14007941} a^{8} - \frac{26624}{14007941} a^{7} + \frac{2664775}{14007941} a^{6} - \frac{209664}{14007941} a^{5} - \frac{1594335}{14007941} a^{4} - \frac{638976}{14007941} a^{3} - \frac{584470}{14007941} a^{2} - \frac{532480}{14007941} a - \frac{1197047}{14007941}$, $\frac{1}{14007941} a^{14} - \frac{4885398}{14007941} a^{9} + \frac{23296}{14007941} a^{8} + \frac{6002702}{14007941} a^{7} + \frac{489216}{14007941} a^{6} - \frac{3563034}{14007941} a^{5} + \frac{3354624}{14007941} a^{4} + \frac{4384}{14007941} a^{3} - \frac{6553221}{14007941} a^{2} + \frac{2311166}{14007941} a + \frac{2555904}{14007941}$, $\frac{1}{14007941} a^{15} + \frac{29120}{14007941} a^{9} + \frac{5307448}{14007941} a^{8} + \frac{698880}{14007941} a^{7} + \frac{711351}{14007941} a^{6} + \frac{5870592}{14007941} a^{5} + \frac{415828}{14007941} a^{4} + \frac{4628859}{14007941} a^{3} + \frac{3134054}{14007941} a^{2} + \frac{1966459}{14007941} a + \frac{3625230}{14007941}$, $\frac{1}{14007941} a^{16} + \frac{4959821}{14007941} a^{9} - \frac{465920}{14007941} a^{8} + \frac{2204720}{14007941} a^{7} + \frac{3571333}{14007941} a^{6} + \frac{4328315}{14007941} a^{5} - \frac{4507495}{14007941} a^{4} - \frac{5936559}{14007941} a^{3} - \frac{2298308}{14007941} a^{2} + \frac{1145259}{14007941} a - \frac{3605996}{14007941}$, $\frac{1}{14007941} a^{17} - \frac{609280}{14007941} a^{9} - \frac{76946}{14007941} a^{8} - \frac{1589627}{14007941} a^{7} + \frac{400873}{14007941} a^{6} + \frac{3600690}{14007941} a^{5} - \frac{6366606}{14007941} a^{4} + \frac{2609312}{14007941} a^{3} + \frac{285165}{14007941} a^{2} + \frac{2283148}{14007941} a - \frac{1956183}{14007941}$

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

Class group and class number

$C_{38836}$, which has order $38836$ (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

\(\Q(\sqrt{-323}) \), 3.3.361.1, 6.0.12165074387.1, \(\Q(\zeta_{19})^+\)

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 $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ $18$ R R $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ $18$

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
17Data not computed
19Data not computed