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Properties

Label 18.0.42858645692...5767.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{45}\cdot 29^{9}$
Root discriminant $83.95$
Ramified primes $3, 29$
Class number $111834$ (GRH)
Class group $[111834]$ (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![1229018617, -719277174, 466948881, -342512940, 444713220, -44037378, 163061514, -2097018, 29950074, -33286, 3090087, 0, 187278, 0, 6615, 0, 126, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 126*x^16 + 6615*x^14 + 187278*x^12 + 3090087*x^10 - 33286*x^9 + 29950074*x^8 - 2097018*x^7 + 163061514*x^6 - 44037378*x^5 + 444713220*x^4 - 342512940*x^3 + 466948881*x^2 - 719277174*x + 1229018617)
 
gp: K = bnfinit(x^18 + 126*x^16 + 6615*x^14 + 187278*x^12 + 3090087*x^10 - 33286*x^9 + 29950074*x^8 - 2097018*x^7 + 163061514*x^6 - 44037378*x^5 + 444713220*x^4 - 342512940*x^3 + 466948881*x^2 - 719277174*x + 1229018617, 1)
 

Normalized defining polynomial

\( x^{18} + 126 x^{16} + 6615 x^{14} + 187278 x^{12} + 3090087 x^{10} - 33286 x^{9} + 29950074 x^{8} - 2097018 x^{7} + 163061514 x^{6} - 44037378 x^{5} + 444713220 x^{4} - 342512940 x^{3} + 466948881 x^{2} - 719277174 x + 1229018617 \)

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:  \(-42858645692297580966155834896045767=-\,3^{45}\cdot 29^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.95$
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:  $3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(783=3^{3}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{783}(1,·)$, $\chi_{783}(260,·)$, $\chi_{783}(262,·)$, $\chi_{783}(521,·)$, $\chi_{783}(523,·)$, $\chi_{783}(782,·)$, $\chi_{783}(86,·)$, $\chi_{783}(88,·)$, $\chi_{783}(347,·)$, $\chi_{783}(349,·)$, $\chi_{783}(608,·)$, $\chi_{783}(610,·)$, $\chi_{783}(173,·)$, $\chi_{783}(175,·)$, $\chi_{783}(434,·)$, $\chi_{783}(436,·)$, $\chi_{783}(695,·)$, $\chi_{783}(697,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6616} a^{9} + \frac{63}{6616} a^{7} + \frac{1323}{6616} a^{5} - \frac{1471}{3308} a^{3} + \frac{1761}{6616} a - \frac{103}{6616}$, $\frac{1}{6616} a^{10} + \frac{63}{6616} a^{8} + \frac{1323}{6616} a^{6} - \frac{1471}{3308} a^{4} + \frac{1761}{6616} a^{2} - \frac{103}{6616} a$, $\frac{1}{6616} a^{11} - \frac{1323}{3308} a^{7} - \frac{283}{6616} a^{5} + \frac{1859}{6616} a^{3} - \frac{103}{6616} a^{2} + \frac{1529}{6616} a - \frac{127}{6616}$, $\frac{1}{6616} a^{12} - \frac{1323}{3308} a^{8} - \frac{283}{6616} a^{6} + \frac{1859}{6616} a^{4} - \frac{103}{6616} a^{3} + \frac{1529}{6616} a^{2} - \frac{127}{6616} a$, $\frac{1}{6616} a^{13} + \frac{1015}{6616} a^{7} + \frac{2653}{6616} a^{5} - \frac{103}{6616} a^{4} - \frac{2587}{6616} a^{3} - \frac{127}{6616} a^{2} + \frac{971}{3308} a - \frac{641}{3308}$, $\frac{1}{7597228242248} a^{14} + \frac{87522263}{7597228242248} a^{13} + \frac{49}{3798614121124} a^{12} - \frac{9206736}{949653530281} a^{11} + \frac{3773}{7597228242248} a^{10} - \frac{140631637}{3798614121124} a^{9} + \frac{36015}{3798614121124} a^{8} + \frac{775437544195}{7597228242248} a^{7} + \frac{352947}{3798614121124} a^{6} + \frac{1813180676311}{3798614121124} a^{5} - \frac{597121106017}{1899307060562} a^{4} + \frac{1837287643209}{3798614121124} a^{3} + \frac{581334088969}{1899307060562} a^{2} - \frac{1983154996163}{7597228242248} a + \frac{101912842902}{949653530281}$, $\frac{1}{7597228242248} a^{15} + \frac{105}{7597228242248} a^{13} + \frac{267827781}{3798614121124} a^{12} + \frac{2205}{3798614121124} a^{11} + \frac{210922491}{7597228242248} a^{10} + \frac{94325}{7597228242248} a^{9} - \frac{1404056500119}{7597228242248} a^{8} + \frac{540225}{3798614121124} a^{7} - \frac{770516019405}{1899307060562} a^{6} + \frac{3176523}{3798614121124} a^{5} - \frac{498354916297}{3798614121124} a^{4} + \frac{1406697939535}{7597228242248} a^{3} + \frac{635045564111}{3798614121124} a^{2} + \frac{3764177132179}{7597228242248} a + \frac{733782403615}{3798614121124}$, $\frac{1}{7597228242248} a^{16} + \frac{532309171}{7597228242248} a^{13} - \frac{735}{949653530281} a^{12} - \frac{46799545}{3798614121124} a^{11} - \frac{37730}{949653530281} a^{10} + \frac{2446521}{3798614121124} a^{9} - \frac{1620675}{1899307060562} a^{8} + \frac{287088052909}{949653530281} a^{7} - \frac{8470728}{949653530281} a^{6} - \frac{720400265215}{1899307060562} a^{5} + \frac{542821876419}{7597228242248} a^{4} + \frac{37882704107}{7597228242248} a^{3} + \frac{1188205826625}{3798614121124} a^{2} - \frac{444443967461}{7597228242248} a - \frac{1212703313583}{3798614121124}$, $\frac{1}{7597228242248} a^{17} - \frac{833}{949653530281} a^{13} + \frac{281213345}{3798614121124} a^{12} - \frac{46648}{949653530281} a^{11} - \frac{482647}{3798614121124} a^{10} - \frac{2244935}{1899307060562} a^{9} - \frac{3701037875557}{7597228242248} a^{8} - \frac{13714512}{949653530281} a^{7} - \frac{449425239311}{7597228242248} a^{6} - \frac{84001386}{949653530281} a^{5} - \frac{91357404849}{949653530281} a^{4} + \frac{715902597307}{7597228242248} a^{3} - \frac{134164402259}{7597228242248} a^{2} + \frac{100134242103}{1899307060562} a + \frac{346470364962}{949653530281}$

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

Class group and class number

$C_{111834}$, which has order $111834$ (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:  \( 40934.0329443 \) (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

\(\Q(\sqrt{-87}) \), \(\Q(\zeta_{9})^+\), 6.0.480048687.1, \(\Q(\zeta_{27})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ $18$ R $18$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ $18$

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
3Data not computed
29Data not computed