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Properties

Label 18.18.1809732592...0000.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 3^{24}\cdot 5^{8}\cdot 29^{3}\cdot 37^{6}$
Root discriminant $103.35$
Ramified primes $2, 3, 5, 29, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times (C_3\times A_4):S_3$ (as 18T156)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17858, -55974, -347892, 626660, 2308524, -857994, -5455849, -3120024, 1419528, 1637522, 150759, -218556, -50868, 10566, 3693, -160, -102, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 102*x^16 - 160*x^15 + 3693*x^14 + 10566*x^13 - 50868*x^12 - 218556*x^11 + 150759*x^10 + 1637522*x^9 + 1419528*x^8 - 3120024*x^7 - 5455849*x^6 - 857994*x^5 + 2308524*x^4 + 626660*x^3 - 347892*x^2 - 55974*x + 17858)
 
gp: K = bnfinit(x^18 - 102*x^16 - 160*x^15 + 3693*x^14 + 10566*x^13 - 50868*x^12 - 218556*x^11 + 150759*x^10 + 1637522*x^9 + 1419528*x^8 - 3120024*x^7 - 5455849*x^6 - 857994*x^5 + 2308524*x^4 + 626660*x^3 - 347892*x^2 - 55974*x + 17858, 1)
 

Normalized defining polynomial

\( x^{18} - 102 x^{16} - 160 x^{15} + 3693 x^{14} + 10566 x^{13} - 50868 x^{12} - 218556 x^{11} + 150759 x^{10} + 1637522 x^{9} + 1419528 x^{8} - 3120024 x^{7} - 5455849 x^{6} - 857994 x^{5} + 2308524 x^{4} + 626660 x^{3} - 347892 x^{2} - 55974 x + 17858 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1809732592884732626462062694400000000=2^{18}\cdot 3^{24}\cdot 5^{8}\cdot 29^{3}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $103.35$
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:  $2, 3, 5, 29, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{76} a^{16} + \frac{3}{76} a^{15} - \frac{9}{76} a^{14} + \frac{2}{19} a^{13} + \frac{4}{19} a^{12} + \frac{7}{19} a^{10} - \frac{5}{38} a^{9} + \frac{17}{76} a^{8} + \frac{35}{76} a^{7} - \frac{25}{76} a^{6} - \frac{1}{2} a^{5} + \frac{5}{38} a^{4} - \frac{1}{2} a^{3} + \frac{7}{19} a^{2} - \frac{7}{19} a - \frac{7}{38}$, $\frac{1}{968066452121470392550530694602547641605270644} a^{17} - \frac{2267388460667815809059884914755341126996503}{968066452121470392550530694602547641605270644} a^{16} + \frac{250106380093077117646961022093818250342809}{968066452121470392550530694602547641605270644} a^{15} + \frac{61238349195462398298025997537686689069443611}{968066452121470392550530694602547641605270644} a^{14} + \frac{96788379468655006511678781483693569123866281}{484033226060735196275265347301273820802635322} a^{13} - \frac{22870422975843069351379447416546783889547647}{242016613030367598137632673650636910401317661} a^{12} - \frac{41859628397505057233159347911157368756117485}{484033226060735196275265347301273820802635322} a^{11} - \frac{215932523908805553650220597062900859748707973}{484033226060735196275265347301273820802635322} a^{10} + \frac{55481837862006637971110411768325349653988899}{968066452121470392550530694602547641605270644} a^{9} - \frac{24378705901837776885148760909917351571370201}{50950865901130020660554247084344612716066876} a^{8} + \frac{391590702727703228860578307804071254777714345}{968066452121470392550530694602547641605270644} a^{7} + \frac{103231585048332541074636850185472494851288807}{968066452121470392550530694602547641605270644} a^{6} + \frac{100086452519915905932859366811593357469747977}{484033226060735196275265347301273820802635322} a^{5} + \frac{35595741390583178005060256601363894597101189}{242016613030367598137632673650636910401317661} a^{4} - \frac{96305595787764782324668108193320203365761415}{242016613030367598137632673650636910401317661} a^{3} - \frac{26955890166245664046915915625896774638254326}{242016613030367598137632673650636910401317661} a^{2} - \frac{20327341309791681126523140888967053849859434}{242016613030367598137632673650636910401317661} a + \frac{226391131216403455381083025715005981968146659}{484033226060735196275265347301273820802635322}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:  \( 11616243686200 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times (C_3\times A_4):S_3$ (as 18T156):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 38 conjugacy class representatives for $C_2\times (C_3\times A_4):S_3$
Character table for $C_2\times (C_3\times A_4):S_3$ is not computed

Intermediate fields

3.3.148.1, 6.6.2540864.1, 9.9.1076763238920000.1

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

Sibling fields

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 R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ R ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$2$2.6.6.8$x^{6} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
2.6.6.8$x^{6} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
2.6.6.8$x^{6} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
3Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.12.8.1$x^{12} - 30 x^{9} + 175 x^{6} + 500 x^{3} + 5000$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$29$29.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
29.6.3.1$x^{6} - 58 x^{4} + 841 x^{2} - 219501$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
$37$37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.6.0.1$x^{6} - x + 20$$1$$6$$0$$C_6$$[\ ]^{6}$