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Properties

Label 18.0.22733694229...1875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,5^{9}\cdot 7^{9}\cdot 19^{16}$
Root discriminant $81.04$
Ramified primes $5, 7, 19$
Class number $120992$ (GRH)
Class group $[2, 4, 15124]$ (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![1741936681, -780172810, 1225758137, -485008616, 405966017, -141089824, 82531870, -25028575, 11308280, -2959970, 1080280, -239315, 71783, -12969, 3187, -434, 85, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 85*x^16 - 434*x^15 + 3187*x^14 - 12969*x^13 + 71783*x^12 - 239315*x^11 + 1080280*x^10 - 2959970*x^9 + 11308280*x^8 - 25028575*x^7 + 82531870*x^6 - 141089824*x^5 + 405966017*x^4 - 485008616*x^3 + 1225758137*x^2 - 780172810*x + 1741936681)
 
gp: K = bnfinit(x^18 - 7*x^17 + 85*x^16 - 434*x^15 + 3187*x^14 - 12969*x^13 + 71783*x^12 - 239315*x^11 + 1080280*x^10 - 2959970*x^9 + 11308280*x^8 - 25028575*x^7 + 82531870*x^6 - 141089824*x^5 + 405966017*x^4 - 485008616*x^3 + 1225758137*x^2 - 780172810*x + 1741936681, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 85 x^{16} - 434 x^{15} + 3187 x^{14} - 12969 x^{13} + 71783 x^{12} - 239315 x^{11} + 1080280 x^{10} - 2959970 x^{9} + 11308280 x^{8} - 25028575 x^{7} + 82531870 x^{6} - 141089824 x^{5} + 405966017 x^{4} - 485008616 x^{3} + 1225758137 x^{2} - 780172810 x + 1741936681 \)

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:  \(-22733694229750493213828467513671875=-\,5^{9}\cdot 7^{9}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.04$
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:  $5, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(665=5\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{665}(384,·)$, $\chi_{665}(1,·)$, $\chi_{665}(386,·)$, $\chi_{665}(454,·)$, $\chi_{665}(139,·)$, $\chi_{665}(524,·)$, $\chi_{665}(594,·)$, $\chi_{665}(596,·)$, $\chi_{665}(349,·)$, $\chi_{665}(351,·)$, $\chi_{665}(419,·)$, $\chi_{665}(36,·)$, $\chi_{665}(104,·)$, $\chi_{665}(106,·)$, $\chi_{665}(491,·)$, $\chi_{665}(176,·)$, $\chi_{665}(244,·)$, $\chi_{665}(631,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1907997756384950051713214142440283663903924457779445265951} a^{17} - \frac{354134399317941959888596544444297065337090477754182355737}{1907997756384950051713214142440283663903924457779445265951} a^{16} - \frac{296408539091144450322839079625435268264275637948046112696}{1907997756384950051713214142440283663903924457779445265951} a^{15} - \frac{721405369925821294827298817713897202398253719253524486096}{1907997756384950051713214142440283663903924457779445265951} a^{14} + \frac{443448280453173461331189612653818183344781346960562344215}{1907997756384950051713214142440283663903924457779445265951} a^{13} + \frac{508470458828930314185275654457175708340301807470436179174}{1907997756384950051713214142440283663903924457779445265951} a^{12} - \frac{769443834828984501257054328522315127548913137792231640397}{1907997756384950051713214142440283663903924457779445265951} a^{11} + \frac{204234814569012167816200010656279206268237384629015612627}{1907997756384950051713214142440283663903924457779445265951} a^{10} - \frac{443608934978345433409231360981025207755289764936815016862}{1907997756384950051713214142440283663903924457779445265951} a^{9} + \frac{212104543182043176844840998049602801831074800813534740076}{1907997756384950051713214142440283663903924457779445265951} a^{8} - \frac{262593501303633903159822278080633103604794424225107995840}{1907997756384950051713214142440283663903924457779445265951} a^{7} + \frac{274492064379572795422433925956252306950932563306804452348}{1907997756384950051713214142440283663903924457779445265951} a^{6} - \frac{408338180843316399083950178405829828767562600121299893100}{1907997756384950051713214142440283663903924457779445265951} a^{5} + \frac{114600936164933209254766879924152892394431074024290314892}{1907997756384950051713214142440283663903924457779445265951} a^{4} - \frac{24315166614371382348846337596987477565370124611865154343}{1907997756384950051713214142440283663903924457779445265951} a^{3} - \frac{516273860768707278249955535317976101136189380896536454776}{1907997756384950051713214142440283663903924457779445265951} a^{2} + \frac{341332712087419405321945497262534901286611820200934200025}{1907997756384950051713214142440283663903924457779445265951} a + \frac{462086411191433028798594839443369294257827491330615918894}{1907997756384950051713214142440283663903924457779445265951}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{15124}$, which has order $120992$ (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

\(\Q(\sqrt{-35}) \), 3.3.361.1, 6.0.5587512875.3, \(\Q(\zeta_{19})^+\)

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 $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ $18$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ $18$

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
5Data not computed
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19Data not computed