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Properties

Label 18.0.12516723252...1339.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 7^{12}\cdot 17^{9}$
Root discriminant $78.40$
Ramified primes $3, 7, 17$
Class number $53144$ (GRH)
Class group $[2, 2, 13286]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![210280141, -45736635, 173222787, -37914480, 75490509, -8508129, 15298582, -907068, 2166219, -201688, 291324, -52812, 32653, -5913, 2178, -299, 75, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 75*x^16 - 299*x^15 + 2178*x^14 - 5913*x^13 + 32653*x^12 - 52812*x^11 + 291324*x^10 - 201688*x^9 + 2166219*x^8 - 907068*x^7 + 15298582*x^6 - 8508129*x^5 + 75490509*x^4 - 37914480*x^3 + 173222787*x^2 - 45736635*x + 210280141)
 
gp: K = bnfinit(x^18 - 6*x^17 + 75*x^16 - 299*x^15 + 2178*x^14 - 5913*x^13 + 32653*x^12 - 52812*x^11 + 291324*x^10 - 201688*x^9 + 2166219*x^8 - 907068*x^7 + 15298582*x^6 - 8508129*x^5 + 75490509*x^4 - 37914480*x^3 + 173222787*x^2 - 45736635*x + 210280141, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 75 x^{16} - 299 x^{15} + 2178 x^{14} - 5913 x^{13} + 32653 x^{12} - 52812 x^{11} + 291324 x^{10} - 201688 x^{9} + 2166219 x^{8} - 907068 x^{7} + 15298582 x^{6} - 8508129 x^{5} + 75490509 x^{4} - 37914480 x^{3} + 173222787 x^{2} - 45736635 x + 210280141 \)

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:  \(-12516723252931349928823131701451339=-\,3^{27}\cdot 7^{12}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.40$
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:  $3, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1071=3^{2}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1071}(256,·)$, $\chi_{1071}(1,·)$, $\chi_{1071}(968,·)$, $\chi_{1071}(970,·)$, $\chi_{1071}(715,·)$, $\chi_{1071}(205,·)$, $\chi_{1071}(919,·)$, $\chi_{1071}(662,·)$, $\chi_{1071}(407,·)$, $\chi_{1071}(611,·)$, $\chi_{1071}(613,·)$, $\chi_{1071}(358,·)$, $\chi_{1071}(50,·)$, $\chi_{1071}(305,·)$, $\chi_{1071}(562,·)$, $\chi_{1071}(1019,·)$, $\chi_{1071}(764,·)$, $\chi_{1071}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{10} - \frac{1}{10} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{2} a^{5} + \frac{1}{10} a^{3} + \frac{1}{10} a^{2} + \frac{3}{10}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{11} - \frac{1}{10} a^{10} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{2} a^{6} + \frac{1}{10} a^{4} + \frac{1}{10} a^{3} + \frac{3}{10} a$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{11} - \frac{1}{10} a^{10} - \frac{1}{10} a^{9} + \frac{2}{5} a^{8} + \frac{1}{10} a^{7} + \frac{2}{5} a^{6} + \frac{3}{10} a^{4} + \frac{1}{10} a^{3} + \frac{1}{5} a^{2} + \frac{1}{10}$, $\frac{1}{2330} a^{15} - \frac{73}{2330} a^{14} - \frac{23}{2330} a^{13} - \frac{7}{2330} a^{12} + \frac{37}{466} a^{11} - \frac{29}{466} a^{10} - \frac{15}{466} a^{9} + \frac{853}{2330} a^{8} + \frac{159}{466} a^{7} - \frac{39}{466} a^{6} + \frac{741}{2330} a^{5} - \frac{175}{466} a^{4} + \frac{212}{1165} a^{3} - \frac{509}{1165} a^{2} - \frac{37}{1165} a - \frac{697}{2330}$, $\frac{1}{2330} a^{16} + \frac{7}{2330} a^{14} - \frac{11}{466} a^{13} - \frac{93}{2330} a^{12} - \frac{77}{1165} a^{11} - \frac{35}{466} a^{10} + \frac{19}{1165} a^{9} + \frac{543}{1165} a^{8} + \frac{289}{2330} a^{7} - \frac{213}{2330} a^{6} + \frac{793}{2330} a^{5} + \frac{391}{2330} a^{4} + \frac{343}{2330} a^{3} + \frac{871}{2330} a^{2} - \frac{74}{233} a + \frac{379}{2330}$, $\frac{1}{11500264136169543417448558551560688415027714812548126483930} a^{17} - \frac{2323573066600543764738241942434817572197038659371966133}{11500264136169543417448558551560688415027714812548126483930} a^{16} - \frac{1804223034955038696631802377125317900837898279088871237}{11500264136169543417448558551560688415027714812548126483930} a^{15} - \frac{108705634000622353864613094137700149811892107330010884959}{11500264136169543417448558551560688415027714812548126483930} a^{14} + \frac{561721165270468772407886547017938251208194938964799367481}{11500264136169543417448558551560688415027714812548126483930} a^{13} + \frac{54742656304936597525381222202577795610782480782361442411}{5750132068084771708724279275780344207513857406274063241965} a^{12} - \frac{223067228926445483283893284356353787594871824430812905407}{2300052827233908683489711710312137683005542962509625296786} a^{11} + \frac{528036572175501484984070379860595550698915798029178889933}{5750132068084771708724279275780344207513857406274063241965} a^{10} + \frac{275496047174968430404802131159948646992383954261195206532}{5750132068084771708724279275780344207513857406274063241965} a^{9} - \frac{5647050522489899361689786621568411996032852739137048669889}{11500264136169543417448558551560688415027714812548126483930} a^{8} - \frac{1398024796892521833336830705541351739783794895525562755419}{11500264136169543417448558551560688415027714812548126483930} a^{7} + \frac{4433045665322614199641457586913115393025450479585576612377}{11500264136169543417448558551560688415027714812548126483930} a^{6} - \frac{3189889397176627251200460241238426254481865642590317970807}{11500264136169543417448558551560688415027714812548126483930} a^{5} + \frac{1328987146653311305788361241688762867327203100371611335932}{5750132068084771708724279275780344207513857406274063241965} a^{4} + \frac{2329716815935392285202479871354711465885695775600625395891}{11500264136169543417448558551560688415027714812548126483930} a^{3} - \frac{2495902965570168180722410354145970706537769380008292570697}{5750132068084771708724279275780344207513857406274063241965} a^{2} - \frac{483477200051296946031458953573323281969993619025612509131}{1150026413616954341744855855156068841502771481254812648393} a + \frac{1554139595187343682421398588341721962328825895575213567451}{11500264136169543417448558551560688415027714812548126483930}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{13286}$, which has order $53144$ (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:  \( 54408.4888887 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-51}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, 6.0.96702579.1, 6.0.232182892179.5, 6.0.318495051.1, 6.0.232182892179.4, 9.9.62523502209.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
7Data not computed
$17$17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$