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Properties

Label 18.12.2052493210...0464.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{12}\cdot 37^{2}\cdot 43^{2}\cdot 101^{9}\cdot 179\cdot 100558321^{2}$
Root discriminant $374.03$
Ramified primes $2, 37, 43, 101, 179, 100558321$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 18T874

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6276416863, 22594393249, 32390759770, 22086498934, 5013215707, -2556688578, -1876953056, -231443086, 142692310, 48261988, -1605105, -2795555, -247265, 69505, 11530, -637, -187, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 187*x^16 - 637*x^15 + 11530*x^14 + 69505*x^13 - 247265*x^12 - 2795555*x^11 - 1605105*x^10 + 48261988*x^9 + 142692310*x^8 - 231443086*x^7 - 1876953056*x^6 - 2556688578*x^5 + 5013215707*x^4 + 22086498934*x^3 + 32390759770*x^2 + 22594393249*x + 6276416863)
 
gp: K = bnfinit(x^18 - 187*x^16 - 637*x^15 + 11530*x^14 + 69505*x^13 - 247265*x^12 - 2795555*x^11 - 1605105*x^10 + 48261988*x^9 + 142692310*x^8 - 231443086*x^7 - 1876953056*x^6 - 2556688578*x^5 + 5013215707*x^4 + 22086498934*x^3 + 32390759770*x^2 + 22594393249*x + 6276416863, 1)
 

Normalized defining polynomial

\( x^{18} - 187 x^{16} - 637 x^{15} + 11530 x^{14} + 69505 x^{13} - 247265 x^{12} - 2795555 x^{11} - 1605105 x^{10} + 48261988 x^{9} + 142692310 x^{8} - 231443086 x^{7} - 1876953056 x^{6} - 2556688578 x^{5} + 5013215707 x^{4} + 22086498934 x^{3} + 32390759770 x^{2} + 22594393249 x + 6276416863 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-20524932102406811680483147938644837890889150464=-\,2^{12}\cdot 37^{2}\cdot 43^{2}\cdot 101^{9}\cdot 179\cdot 100558321^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $374.03$
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:  $2, 37, 43, 101, 179, 100558321$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{10} a^{15} - \frac{1}{10} a^{14} + \frac{3}{10} a^{13} - \frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{3}{10} a^{9} + \frac{2}{5} a^{7} - \frac{1}{2} a^{6} + \frac{1}{10} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{10} a^{2} - \frac{2}{5} a + \frac{1}{10}$, $\frac{1}{60} a^{16} - \frac{1}{30} a^{15} + \frac{1}{15} a^{14} - \frac{1}{12} a^{13} - \frac{1}{10} a^{12} - \frac{4}{15} a^{11} - \frac{1}{20} a^{10} + \frac{9}{20} a^{9} + \frac{2}{5} a^{8} - \frac{29}{60} a^{7} - \frac{1}{15} a^{6} - \frac{5}{12} a^{5} - \frac{1}{15} a^{4} - \frac{11}{60} a^{3} + \frac{5}{12} a^{2} + \frac{1}{4} a + \frac{29}{60}$, $\frac{1}{77948578595648659575645623635160237976692872773202260} a^{17} + \frac{64008237572966929216668095878158042644804257746961}{77948578595648659575645623635160237976692872773202260} a^{16} + \frac{536592897317036480485573452105921722693272251629}{32779049030970840864443071335222976440997843891170} a^{15} + \frac{32904694705781549285683942100741006025488385578516383}{77948578595648659575645623635160237976692872773202260} a^{14} - \frac{2395326025006841303903590203707868843513265254728463}{25982859531882886525215207878386745992230957591067420} a^{13} - \frac{14981591271434760194277007066387740807942531238902851}{38974289297824329787822811817580118988346436386601130} a^{12} + \frac{2727897454817887146186781538118278195482529064617}{387803873610192336197241908632637999884044143150260} a^{11} + \frac{1375956679096208166951629164270602125592023511280347}{12991429765941443262607603939193372996115478795533710} a^{10} + \frac{88434554450262255347148482535175416000980249758939}{25982859531882886525215207878386745992230957591067420} a^{9} + \frac{171466144915879224630856147628181961686345209470369}{1163411620830577008591725725897913999652132429450780} a^{8} + \frac{1760166579528998609093220616443600985101766994686237}{15589715719129731915129124727032047595338574554640452} a^{7} + \frac{32403387593103517303886033478990263138656335753519203}{77948578595648659575645623635160237976692872773202260} a^{6} + \frac{5162895149875334697323791502856480729324506249789941}{15589715719129731915129124727032047595338574554640452} a^{5} - \frac{28882781332344381258453815289241206280773255678484739}{77948578595648659575645623635160237976692872773202260} a^{4} + \frac{308184406888364600594449769917702148366600179599059}{3897428929782432978782281181758011898834643638660113} a^{3} - \frac{4452831294445791184623977284064494022727516960792271}{12991429765941443262607603939193372996115478795533710} a^{2} + \frac{18958041459788468710243582624858862771704282290610879}{38974289297824329787822811817580118988346436386601130} a + \frac{5618212986369146277104145894063053191582467119730937}{25982859531882886525215207878386745992230957591067420}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $14$
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:  \( 30599256226100000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T874:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 279936
The 174 conjugacy class representatives for t18n874 are not computed
Character table for t18n874 is not computed

Intermediate fields

\(\Q(\sqrt{101}) \), 3.3.404.1 x3, 6.6.16484816.1

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

Sibling fields

Degree 18 siblings: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.3.2.2$x^{3} + 74$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
43.3.2.2$x^{3} + 387$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
101Data not computed
$179$$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
179.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
100558321Data not computed