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Properties

Label 18.0.13719535328...2944.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{31}\cdot 13^{9}\cdot 19^{14}$
Root discriminant $472.39$
Ramified primes $2, 3, 13, 19$
Class number $8697190320$ (GRH)
Class group $[2, 6, 724765860]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![380974814208, -307300490496, 211735347840, -89371535328, 34090131744, -9783568800, 2645917704, -548111610, 105217326, -14553153, 2697543, -556254, 145350, -18684, 576, 408, -36, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 36*x^16 + 408*x^15 + 576*x^14 - 18684*x^13 + 145350*x^12 - 556254*x^11 + 2697543*x^10 - 14553153*x^9 + 105217326*x^8 - 548111610*x^7 + 2645917704*x^6 - 9783568800*x^5 + 34090131744*x^4 - 89371535328*x^3 + 211735347840*x^2 - 307300490496*x + 380974814208)
 
gp: K = bnfinit(x^18 - 3*x^17 - 36*x^16 + 408*x^15 + 576*x^14 - 18684*x^13 + 145350*x^12 - 556254*x^11 + 2697543*x^10 - 14553153*x^9 + 105217326*x^8 - 548111610*x^7 + 2645917704*x^6 - 9783568800*x^5 + 34090131744*x^4 - 89371535328*x^3 + 211735347840*x^2 - 307300490496*x + 380974814208, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 36 x^{16} + 408 x^{15} + 576 x^{14} - 18684 x^{13} + 145350 x^{12} - 556254 x^{11} + 2697543 x^{10} - 14553153 x^{9} + 105217326 x^{8} - 548111610 x^{7} + 2645917704 x^{6} - 9783568800 x^{5} + 34090131744 x^{4} - 89371535328 x^{3} + 211735347840 x^{2} - 307300490496 x + 380974814208 \)

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:  \(-1371953532870850784692114650282846158453984722944=-\,2^{18}\cdot 3^{31}\cdot 13^{9}\cdot 19^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $472.39$
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:  $2, 3, 13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{6} - \frac{1}{12} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{12} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{24} a^{8} - \frac{1}{12} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{216} a^{9} + \frac{1}{72} a^{8} + \frac{1}{36} a^{6} - \frac{1}{24} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{432} a^{10} - \frac{1}{432} a^{9} + \frac{1}{72} a^{8} + \frac{1}{72} a^{7} + \frac{1}{144} a^{6} + \frac{1}{48} a^{5} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{432} a^{11} - \frac{1}{432} a^{9} - \frac{1}{72} a^{8} + \frac{1}{48} a^{7} + \frac{1}{36} a^{6} + \frac{1}{48} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{1}{12} a^{2} + \frac{1}{3}$, $\frac{1}{1728} a^{12} - \frac{1}{864} a^{11} - \frac{1}{1728} a^{10} + \frac{11}{576} a^{8} - \frac{7}{288} a^{7} + \frac{5}{192} a^{6} - \frac{1}{16} a^{5} + \frac{7}{32} a^{4} + \frac{7}{24} a^{3} - \frac{5}{24} a^{2} + \frac{1}{3} a$, $\frac{1}{1728} a^{13} - \frac{1}{1728} a^{11} - \frac{1}{864} a^{10} - \frac{1}{576} a^{9} - \frac{1}{72} a^{8} - \frac{1}{576} a^{7} - \frac{1}{96} a^{6} - \frac{5}{96} a^{5} - \frac{1}{48} a^{4} - \frac{5}{12} a^{2} - \frac{1}{3} a$, $\frac{1}{10368} a^{14} - \frac{1}{3456} a^{13} - \frac{1}{3456} a^{12} + \frac{1}{1152} a^{11} - \frac{1}{3456} a^{10} - \frac{1}{3456} a^{9} - \frac{5}{1152} a^{8} + \frac{7}{1152} a^{7} + \frac{5}{576} a^{6} - \frac{5}{576} a^{5} + \frac{1}{48} a^{4} - \frac{7}{48} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{51840} a^{15} + \frac{1}{25920} a^{14} + \frac{1}{8640} a^{13} + \frac{1}{960} a^{11} + \frac{1}{2160} a^{10} + \frac{1}{480} a^{9} + \frac{1}{1440} a^{8} - \frac{71}{5760} a^{7} + \frac{91}{2880} a^{6} - \frac{193}{2880} a^{5} - \frac{29}{160} a^{4} + \frac{11}{48} a^{3} - \frac{17}{120} a^{2} + \frac{7}{30} a - \frac{2}{5}$, $\frac{1}{207360} a^{16} + \frac{1}{103680} a^{14} - \frac{7}{34560} a^{13} + \frac{1}{3840} a^{12} - \frac{29}{34560} a^{11} - \frac{1}{1728} a^{10} + \frac{1}{768} a^{9} - \frac{319}{23040} a^{8} + \frac{3}{1280} a^{7} - \frac{5}{256} a^{6} + \frac{157}{5760} a^{5} - \frac{71}{320} a^{4} + \frac{21}{160} a^{3} + \frac{1}{15} a^{2} - \frac{13}{60} a + \frac{1}{5}$, $\frac{1}{3210556248495035035011627095657545597660957011809280} a^{17} + \frac{4351956907214482399534425319412701094331170131}{3210556248495035035011627095657545597660957011809280} a^{16} - \frac{988656723591066405728062130422462238990766493}{107018541616501167833720903188584853255365233726976} a^{15} - \frac{22134714095816728039785599976910033923255792421}{802639062123758758752906773914386399415239252952320} a^{14} - \frac{2663920657122348440351474748200064270044707805}{26754635404125291958430225797146213313841308431744} a^{13} - \frac{11602339820774941960315730987890852489124116693}{53509270808250583916860451594292426627682616863488} a^{12} + \frac{29401309316739059099341681351027267579344532757}{59454745342500648796511612882547140697425129848320} a^{11} + \frac{5057789742429477149092264043344301446522797937}{535092708082505839168604515942924266276826168634880} a^{10} - \frac{18107330386771426336780574250635909330438171171}{15509933567608864903437812056316645399328294743040} a^{9} - \frac{45227496415315550581557754737263548591630964701}{118909490685001297593023225765094281394850259696640} a^{8} - \frac{19340518966501969579871635076750970693902061991}{14863686335625162199127903220636785174356282462080} a^{7} - \frac{2128068909902766889310991559469371476382843285051}{59454745342500648796511612882547140697425129848320} a^{6} + \frac{2023602232525608499241370861924849064125907748797}{89182118013750973194767419323820711046137694772480} a^{5} - \frac{122466972506991297428610135551887704743516761}{5535823588687211247347449989063979580765840768} a^{4} + \frac{508454658219653056816590493208067098271903820573}{1486368633562516219912790322063678517435628246208} a^{3} + \frac{10708052071733983080932178094799036301373379081}{232245098994143159361373487822449768349316913470} a^{2} - \frac{364883597435615985293380484936263640904093627103}{928980395976572637445493951289799073397267653880} a - \frac{34999624669675137713236163063406245771009882563}{77415032998047719787124495940816589449772304490}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{724765860}$, which has order $8697190320$ (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:  \( 15773424688.63964 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-39}) \), 3.3.4104.1, 3.3.29241.1, 6.0.5635542809871.3, 6.0.111011000256.3, 9.9.6566954215853707776.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 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 R R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
3Data not computed
$13$13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.6.4.3$x^{6} + 95 x^{3} + 2888$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.12.10.1$x^{12} - 171 x^{6} + 23104$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$