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Properties

Label 18.0.12832361367...6544.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{45}\cdot 13^{9}$
Root discriminant $89.22$
Ramified primes $2, 3, 13$
Class number $72$ (GRH)
Class group $[2, 36]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![666832, -2485872, 4302216, -5016612, 3698316, -1532268, 890862, -567216, 161568, -47605, 26370, -2394, 4281, 828, 414, -21, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 21*x^15 + 414*x^14 + 828*x^13 + 4281*x^12 - 2394*x^11 + 26370*x^10 - 47605*x^9 + 161568*x^8 - 567216*x^7 + 890862*x^6 - 1532268*x^5 + 3698316*x^4 - 5016612*x^3 + 4302216*x^2 - 2485872*x + 666832)
 
gp: K = bnfinit(x^18 - 21*x^15 + 414*x^14 + 828*x^13 + 4281*x^12 - 2394*x^11 + 26370*x^10 - 47605*x^9 + 161568*x^8 - 567216*x^7 + 890862*x^6 - 1532268*x^5 + 3698316*x^4 - 5016612*x^3 + 4302216*x^2 - 2485872*x + 666832, 1)
 

Normalized defining polynomial

\( x^{18} - 21 x^{15} + 414 x^{14} + 828 x^{13} + 4281 x^{12} - 2394 x^{11} + 26370 x^{10} - 47605 x^{9} + 161568 x^{8} - 567216 x^{7} + 890862 x^{6} - 1532268 x^{5} + 3698316 x^{4} - 5016612 x^{3} + 4302216 x^{2} - 2485872 x + 666832 \)

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:  \(-128323613672506363698737006390636544=-\,2^{12}\cdot 3^{45}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.22$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{2} a^{6} + \frac{1}{6} a^{3} + \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{10} - \frac{1}{2} a^{7} + \frac{1}{6} a^{4} + \frac{1}{3} a$, $\frac{1}{18} a^{14} - \frac{1}{18} a^{13} + \frac{1}{18} a^{12} + \frac{1}{18} a^{11} - \frac{1}{18} a^{10} + \frac{1}{18} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{18} a^{5} - \frac{1}{18} a^{4} + \frac{1}{18} a^{3} + \frac{2}{9} a^{2} - \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{36} a^{15} - \frac{1}{36} a^{12} - \frac{1}{6} a^{10} - \frac{5}{36} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{7}{36} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{4}{9} a^{3} + \frac{1}{3} a - \frac{2}{9}$, $\frac{1}{32436} a^{16} - \frac{389}{32436} a^{15} - \frac{145}{8109} a^{14} - \frac{749}{10812} a^{13} + \frac{1525}{32436} a^{12} - \frac{505}{8109} a^{11} + \frac{3365}{32436} a^{10} - \frac{3}{212} a^{9} - \frac{539}{2703} a^{8} - \frac{7025}{32436} a^{7} + \frac{11311}{32436} a^{6} + \frac{4849}{16218} a^{5} + \frac{1129}{5406} a^{4} + \frac{3491}{16218} a^{3} - \frac{2029}{8109} a^{2} - \frac{25}{8109} a + \frac{244}{2703}$, $\frac{1}{867227375085663237038446554691648781719704784488} a^{17} + \frac{963756319359058556765349024400907794909595}{433613687542831618519223277345824390859852392244} a^{16} - \frac{3459819141951752799720341090460630113751171667}{433613687542831618519223277345824390859852392244} a^{15} - \frac{4435056546006651719574639772791108423086066377}{867227375085663237038446554691648781719704784488} a^{14} - \frac{4175915614318970825355179749027646697249935174}{108403421885707904629805819336456097714963098061} a^{13} - \frac{5938126617773107861054930615824196413681295469}{433613687542831618519223277345824390859852392244} a^{12} - \frac{71983751415574547945596892227324087492185606563}{867227375085663237038446554691648781719704784488} a^{11} + \frac{6184376993205787639756916955990953634311416979}{108403421885707904629805819336456097714963098061} a^{10} - \frac{10686927324562267369562583109298608061934351185}{216806843771415809259611638672912195429926196122} a^{9} - \frac{142775775938477208298864956814024112371865171105}{867227375085663237038446554691648781719704784488} a^{8} - \frac{139618596562132161836726561390720716825391571697}{433613687542831618519223277345824390859852392244} a^{7} - \frac{90118236110934275441130647342479631953201090621}{433613687542831618519223277345824390859852392244} a^{6} - \frac{68315817889779287863426439776166012449372671773}{433613687542831618519223277345824390859852392244} a^{5} + \frac{51573627664843060655937148840001203973032861883}{216806843771415809259611638672912195429926196122} a^{4} + \frac{51394949526491277821290603479494691203886268571}{216806843771415809259611638672912195429926196122} a^{3} + \frac{31428597754263577812781291374096010033419539797}{216806843771415809259611638672912195429926196122} a^{2} - \frac{19678024289287876494035698065522271204638680371}{108403421885707904629805819336456097714963098061} a - \frac{28812351742738465613357404624560064892500022077}{108403421885707904629805819336456097714963098061}$

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

Class group and class number

$C_{2}\times C_{36}$, which has order $72$ (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:  \( 2199571857.9102173 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{-39}) \), 3.1.243.1, 6.0.389191959.7, 9.1.2008387814976.4

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
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}$