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Properties

Label 18.0.11990701314...5739.5
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 59^{9}$
Root discriminant $28.11$
Ramified primes $7, 59$
Class number $9$
Class group $[9]$
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28133, -12418, 11816, 9199, -6368, 5638, -3383, -2446, 2630, 1476, 280, 18, 107, -48, 68, -6, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 4*x^16 - 6*x^15 + 68*x^14 - 48*x^13 + 107*x^12 + 18*x^11 + 280*x^10 + 1476*x^9 + 2630*x^8 - 2446*x^7 - 3383*x^6 + 5638*x^5 - 6368*x^4 + 9199*x^3 + 11816*x^2 - 12418*x + 28133)
 
gp: K = bnfinit(x^18 - 2*x^17 + 4*x^16 - 6*x^15 + 68*x^14 - 48*x^13 + 107*x^12 + 18*x^11 + 280*x^10 + 1476*x^9 + 2630*x^8 - 2446*x^7 - 3383*x^6 + 5638*x^5 - 6368*x^4 + 9199*x^3 + 11816*x^2 - 12418*x + 28133, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 4 x^{16} - 6 x^{15} + 68 x^{14} - 48 x^{13} + 107 x^{12} + 18 x^{11} + 280 x^{10} + 1476 x^{9} + 2630 x^{8} - 2446 x^{7} - 3383 x^{6} + 5638 x^{5} - 6368 x^{4} + 9199 x^{3} + 11816 x^{2} - 12418 x + 28133 \)

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:  \(-119907013147065124216135739=-\,7^{12}\cdot 59^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.11$
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:  $7, 59$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{10} - \frac{2}{7} a^{9} + \frac{1}{7} a^{8} - \frac{1}{7} a^{7} + \frac{2}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{2}{7} a^{3}$, $\frac{1}{7} a^{12} - \frac{3}{7} a^{10} - \frac{1}{7} a^{9} + \frac{1}{7} a^{7} + \frac{1}{7} a^{6} - \frac{2}{7} a^{5} + \frac{1}{7} a^{4} + \frac{2}{7} a^{3}$, $\frac{1}{7} a^{13} + \frac{3}{7} a^{10} + \frac{1}{7} a^{9} - \frac{3}{7} a^{8} - \frac{2}{7} a^{7} - \frac{3}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} - \frac{1}{7} a^{3}$, $\frac{1}{49} a^{14} - \frac{1}{49} a^{13} + \frac{1}{49} a^{11} + \frac{2}{7} a^{10} - \frac{8}{49} a^{8} - \frac{20}{49} a^{7} - \frac{10}{49} a^{6} + \frac{10}{49} a^{5} + \frac{16}{49} a^{4} - \frac{24}{49} a^{3} - \frac{3}{7} a^{2} + \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{5145} a^{15} - \frac{1}{1715} a^{14} - \frac{74}{1715} a^{13} + \frac{38}{1029} a^{12} - \frac{13}{1029} a^{11} + \frac{229}{735} a^{10} - \frac{1541}{5145} a^{9} - \frac{193}{5145} a^{8} + \frac{548}{5145} a^{7} + \frac{66}{1715} a^{6} + \frac{1319}{5145} a^{5} - \frac{43}{735} a^{4} - \frac{54}{1715} a^{3} + \frac{226}{735} a^{2} + \frac{173}{735} a + \frac{118}{735}$, $\frac{1}{117516945} a^{16} - \frac{841}{23503389} a^{15} + \frac{7611}{3013255} a^{14} + \frac{6802009}{117516945} a^{13} + \frac{586483}{23503389} a^{12} + \frac{485678}{117516945} a^{11} + \frac{36155783}{117516945} a^{10} - \frac{19556232}{39172315} a^{9} + \frac{19685599}{117516945} a^{8} - \frac{5195458}{117516945} a^{7} + \frac{31108013}{117516945} a^{6} + \frac{44579116}{117516945} a^{5} - \frac{1430516}{23503389} a^{4} - \frac{390833}{9039765} a^{3} - \frac{2163793}{5596045} a^{2} - \frac{882706}{5596045} a - \frac{6490496}{16788135}$, $\frac{1}{3977398194609857872621006695} a^{17} - \frac{61912817521211106}{37879982805808170215438159} a^{16} - \frac{336308751737290836283027}{3977398194609857872621006695} a^{15} - \frac{435771600040725877167818}{3977398194609857872621006695} a^{14} + \frac{256596836239821401186472577}{3977398194609857872621006695} a^{13} + \frac{88455157873413707570247946}{1325799398203285957540335565} a^{12} - \frac{30723736390136503975281534}{1325799398203285957540335565} a^{11} - \frac{96457010627494103229024467}{568199742087122553231572385} a^{10} - \frac{12885390646910757244033567}{113639948417424510646314477} a^{9} + \frac{247584217019097120690131468}{795479638921971574524201339} a^{8} + \frac{6864893113422134056452630}{265159879640657191508067113} a^{7} + \frac{1254811944016596998959922978}{3977398194609857872621006695} a^{6} + \frac{52576629248686849187622667}{305953707277681374817000515} a^{5} + \frac{1410807590601759908031462377}{3977398194609857872621006695} a^{4} - \frac{8787309152908991384252416}{3977398194609857872621006695} a^{3} - \frac{74986939408609429672327873}{189399914029040851077190795} a^{2} - \frac{158742961646454070792698404}{568199742087122553231572385} a - \frac{228309377134080425824723513}{568199742087122553231572385}$

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

Class group and class number

$C_{9}$, which has order $9$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 73451.9598139 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-59}) \), 3.1.2891.2 x3, \(\Q(\zeta_{7})^+\), 6.0.493114979.1, 6.0.493114979.4, 6.0.10063571.1 x2, 9.3.24162633971.4 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 6 sibling: 6.0.10063571.1
Degree 9 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ R

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$59$59.6.3.2$x^{6} - 3481 x^{2} + 3491443$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
59.6.3.2$x^{6} - 3481 x^{2} + 3491443$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
59.6.3.2$x^{6} - 3481 x^{2} + 3491443$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$