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Properties

Label 18.0.30032008214...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 5^{9}\cdot 7^{15}\cdot 13^{15}\cdot 17^{6}$
Root discriminant $493.40$
Ramified primes $2, 5, 7, 13, 17$
Class number $5672242080$ (GRH)
Class group $[2, 6, 472686840]$ (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![29398869968896, -7002971574528, 6364008129664, 230802504368, 161608733256, -42706675732, 8509113946, 176088726, 93538987, -3787964, -1687355, -856236, -79910, 26100, 6112, -186, -77, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 77*x^16 - 186*x^15 + 6112*x^14 + 26100*x^13 - 79910*x^12 - 856236*x^11 - 1687355*x^10 - 3787964*x^9 + 93538987*x^8 + 176088726*x^7 + 8509113946*x^6 - 42706675732*x^5 + 161608733256*x^4 + 230802504368*x^3 + 6364008129664*x^2 - 7002971574528*x + 29398869968896)
 
gp: K = bnfinit(x^18 - 4*x^17 - 77*x^16 - 186*x^15 + 6112*x^14 + 26100*x^13 - 79910*x^12 - 856236*x^11 - 1687355*x^10 - 3787964*x^9 + 93538987*x^8 + 176088726*x^7 + 8509113946*x^6 - 42706675732*x^5 + 161608733256*x^4 + 230802504368*x^3 + 6364008129664*x^2 - 7002971574528*x + 29398869968896, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 77 x^{16} - 186 x^{15} + 6112 x^{14} + 26100 x^{13} - 79910 x^{12} - 856236 x^{11} - 1687355 x^{10} - 3787964 x^{9} + 93538987 x^{8} + 176088726 x^{7} + 8509113946 x^{6} - 42706675732 x^{5} + 161608733256 x^{4} + 230802504368 x^{3} + 6364008129664 x^{2} - 7002971574528 x + 29398869968896 \)

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:  \(-3003200821410347323073992859276234352128000000000=-\,2^{18}\cdot 5^{9}\cdot 7^{15}\cdot 13^{15}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $493.40$
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:  $2, 5, 7, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{56} a^{9} - \frac{1}{28} a^{8} - \frac{1}{28} a^{7} - \frac{3}{28} a^{6} - \frac{3}{56} a^{5} + \frac{3}{14} a^{4} + \frac{1}{7} a^{3} - \frac{1}{14} a^{2} - \frac{1}{14} a - \frac{1}{7}$, $\frac{1}{112} a^{10} + \frac{1}{112} a^{8} - \frac{5}{56} a^{7} + \frac{13}{112} a^{6} - \frac{1}{14} a^{5} + \frac{11}{112} a^{4} + \frac{27}{56} a^{3} + \frac{15}{56} a^{2} - \frac{11}{28} a - \frac{1}{7}$, $\frac{1}{224} a^{11} - \frac{1}{224} a^{9} + \frac{1}{28} a^{8} + \frac{17}{224} a^{7} - \frac{3}{28} a^{6} - \frac{39}{224} a^{5} + \frac{1}{14} a^{4} - \frac{7}{16} a^{3} - \frac{2}{7} a^{2} - \frac{1}{28} a - \frac{3}{7}$, $\frac{1}{224} a^{12} - \frac{1}{224} a^{10} + \frac{5}{224} a^{8} - \frac{1}{28} a^{7} + \frac{9}{224} a^{6} - \frac{1}{14} a^{5} + \frac{1}{112} a^{4} + \frac{5}{28} a^{3} + \frac{5}{14} a^{2} + \frac{3}{14} a + \frac{2}{7}$, $\frac{1}{15232} a^{13} + \frac{9}{15232} a^{12} - \frac{5}{15232} a^{11} - \frac{13}{15232} a^{10} - \frac{47}{15232} a^{9} - \frac{839}{15232} a^{8} + \frac{147}{2176} a^{7} + \frac{1789}{15232} a^{6} + \frac{1395}{7616} a^{5} + \frac{59}{7616} a^{4} + \frac{87}{272} a^{3} - \frac{741}{1904} a^{2} - \frac{1}{14} a - \frac{2}{7}$, $\frac{1}{15232} a^{14} - \frac{9}{7616} a^{12} + \frac{1}{476} a^{11} + \frac{1}{7616} a^{10} + \frac{1}{119} a^{9} + \frac{27}{1904} a^{8} - \frac{93}{952} a^{7} - \frac{43}{896} a^{6} + \frac{43}{238} a^{5} + \frac{1429}{7616} a^{4} - \frac{17}{56} a^{3} - \frac{675}{1904} a^{2} + \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{60928} a^{15} - \frac{1}{30464} a^{14} - \frac{1}{30464} a^{13} - \frac{1}{476} a^{12} - \frac{3}{30464} a^{11} - \frac{1}{3808} a^{10} - \frac{3}{476} a^{9} + \frac{431}{15232} a^{8} + \frac{2525}{60928} a^{7} + \frac{2241}{30464} a^{6} - \frac{5319}{30464} a^{5} + \frac{97}{15232} a^{4} - \frac{145}{7616} a^{3} + \frac{11}{32} a^{2} - \frac{13}{56} a$, $\frac{1}{3594752} a^{16} - \frac{19}{3594752} a^{15} - \frac{23}{898688} a^{14} + \frac{25}{1797376} a^{13} - \frac{545}{256768} a^{12} + \frac{2595}{1797376} a^{11} - \frac{1809}{449344} a^{10} - \frac{3053}{898688} a^{9} + \frac{26263}{513536} a^{8} + \frac{189797}{3594752} a^{7} + \frac{107857}{898688} a^{6} - \frac{206615}{1797376} a^{5} + \frac{1001}{128384} a^{4} + \frac{5739}{449344} a^{3} + \frac{54237}{224672} a^{2} - \frac{645}{3304} a + \frac{96}{413}$, $\frac{1}{25479797930897490563124383686711587084776897416425294837705586711699408661417984} a^{17} + \frac{2772249952864713070788988035388252904992393944397745100517617277154538739}{25479797930897490563124383686711587084776897416425294837705586711699408661417984} a^{16} - \frac{295525643852885637997102248658289231427196123509296994540110086148305163}{187351455374246254140620468284644022682183069238421285571364608174260357804544} a^{15} - \frac{45199458211635571132160389974429007881398551774461458439617332825133713985}{12739898965448745281562191843355793542388448708212647418852793355849704330708992} a^{14} + \frac{53767120096636247712226768020437183235740160452084278178402953697833793321}{12739898965448745281562191843355793542388448708212647418852793355849704330708992} a^{13} - \frac{7740273282001269067815309976973320051531396272058916248131423851123624504791}{12739898965448745281562191843355793542388448708212647418852793355849704330708992} a^{12} + \frac{5215543548029683517325691305903381241660785403382485868932556912229145645735}{3184974741362186320390547960838948385597112177053161854713198338962426082677248} a^{11} - \frac{393176386763990598364661105193339549936121951092563072435532629474205562491}{107965245469904621030188066469116894427020751764513961176718587761438172294144} a^{10} - \frac{198224266972809192470415151251629845897334954734113978280900526163583773602295}{25479797930897490563124383686711587084776897416425294837705586711699408661417984} a^{9} - \frac{164702408242625602446264829102483569140436496055892061916995014669898649310221}{25479797930897490563124383686711587084776897416425294837705586711699408661417984} a^{8} + \frac{82829093803118954796623900377650687361718821221135158528599320411760509621281}{796243685340546580097636990209737096399278044263290463678299584740606520669312} a^{7} + \frac{1462126666395301446407893846493460804179298510197762235675417528679198154666443}{12739898965448745281562191843355793542388448708212647418852793355849704330708992} a^{6} + \frac{958347723961682193209594461369954635091454814714052257366771112916058892036125}{6369949482724372640781095921677896771194224354106323709426396677924852165354496} a^{5} + \frac{593933079314837404163243776324484086413782640936483451364777220116987308341715}{3184974741362186320390547960838948385597112177053161854713198338962426082677248} a^{4} - \frac{73993164129788638730180128466532141561333248336178789884437148756121568883}{13382246812446161010044319163188858763013076374172948969383186298161454128896} a^{3} - \frac{2104591899127602352974451797188451711968840453108499865823749718443113711827}{11709465960890390883788779267790251417636441827401330348210288010891272362784} a^{2} + \frac{143968535796356409502398125220095644253403905476053144722607528911552556255}{344396057673246790699669978464419159342248289041215598476773176790919775376} a - \frac{2806777338456219850350857462439441988563203917693852114981458130933010833}{6149929601307978405351249615436056416825862304307421401370949585552138846}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{472686840}$, which has order $5672242080$ (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:  \( 55972376075.29769 \) (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

\(\Q(\sqrt{-455}) \), 3.3.114920.1, 3.3.8281.1, Deg 6, 6.0.780040181375.1, Deg 9

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$$0$Trivial$[\ ]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed
13Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$