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Properties

Label 18.0.31978065643...5216.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 37^{14}\cdot 59^{9}$
Root discriminant $202.22$
Ramified primes $2, 37, 59$
Class number $23378544$ (GRH)
Class group $[6, 6, 649404]$ (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![166746964831, -77705750398, 80685361261, -27720649061, 16263505093, -4387355467, 1882145661, -411169037, 142410717, -25521676, 7509688, -1103801, 283423, -33625, 7559, -682, 129, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 129*x^16 - 682*x^15 + 7559*x^14 - 33625*x^13 + 283423*x^12 - 1103801*x^11 + 7509688*x^10 - 25521676*x^9 + 142410717*x^8 - 411169037*x^7 + 1882145661*x^6 - 4387355467*x^5 + 16263505093*x^4 - 27720649061*x^3 + 80685361261*x^2 - 77705750398*x + 166746964831)
 
gp: K = bnfinit(x^18 - 7*x^17 + 129*x^16 - 682*x^15 + 7559*x^14 - 33625*x^13 + 283423*x^12 - 1103801*x^11 + 7509688*x^10 - 25521676*x^9 + 142410717*x^8 - 411169037*x^7 + 1882145661*x^6 - 4387355467*x^5 + 16263505093*x^4 - 27720649061*x^3 + 80685361261*x^2 - 77705750398*x + 166746964831, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 129 x^{16} - 682 x^{15} + 7559 x^{14} - 33625 x^{13} + 283423 x^{12} - 1103801 x^{11} + 7509688 x^{10} - 25521676 x^{9} + 142410717 x^{8} - 411169037 x^{7} + 1882145661 x^{6} - 4387355467 x^{5} + 16263505093 x^{4} - 27720649061 x^{3} + 80685361261 x^{2} - 77705750398 x + 166746964831 \)

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:  \(-319780656433835079525248091919941689225216=-\,2^{12}\cdot 37^{14}\cdot 59^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $202.22$
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:  $2, 37, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{878770834132281201719097847525009748434394727811434726829033701} a^{17} - \frac{98629557173111968806383598396377161557939939895953209030476195}{878770834132281201719097847525009748434394727811434726829033701} a^{16} - \frac{143256474065506441484471994055451528561266411875161138761888330}{878770834132281201719097847525009748434394727811434726829033701} a^{15} + \frac{28298498230677332964688025627743155262489152015799764274680245}{878770834132281201719097847525009748434394727811434726829033701} a^{14} + \frac{28736709950624738937076416599648013939065507938945497670904450}{292923611377427067239699282508336582811464909270478242276344567} a^{13} + \frac{135220630875850233349613738376805920105153555100056374388186467}{878770834132281201719097847525009748434394727811434726829033701} a^{12} - \frac{109840777647341666863746491894408213531543374513450470639013636}{292923611377427067239699282508336582811464909270478242276344567} a^{11} + \frac{431538868720782523190583913451914440601202348702681089079179520}{878770834132281201719097847525009748434394727811434726829033701} a^{10} + \frac{363972872327317323944976123583529994984332378909220126945600656}{878770834132281201719097847525009748434394727811434726829033701} a^{9} + \frac{106267729429662549119788571714734087722287740539712367183698797}{292923611377427067239699282508336582811464909270478242276344567} a^{8} - \frac{78509493842100055895093956017100520249419942002590966209212224}{292923611377427067239699282508336582811464909270478242276344567} a^{7} - \frac{120601101731043799304972986221098852129815377998942917727856241}{878770834132281201719097847525009748434394727811434726829033701} a^{6} + \frac{5493552579777158799188976488420245488486554346637421772232066}{292923611377427067239699282508336582811464909270478242276344567} a^{5} - \frac{364752382659071507959287355436294913401867479195910156771643436}{878770834132281201719097847525009748434394727811434726829033701} a^{4} + \frac{727863090596764436201669709723932024506935479519324436350101}{30302442556285558679968891293965853394289473372808094028587369} a^{3} + \frac{326488530229200805996609709939834474077818664763555613344480461}{878770834132281201719097847525009748434394727811434726829033701} a^{2} + \frac{280899374055589005115414287516825100087590320966021553099745505}{878770834132281201719097847525009748434394727811434726829033701} a + \frac{3870783759841862630791883736142550466732221722935692368859423}{30302442556285558679968891293965853394289473372808094028587369}$

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

Class group and class number

$C_{6}\times C_{6}\times C_{649404}$, which has order $23378544$ (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:  \( 615797.1340659427 \) (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

\(\Q(\sqrt{-59}) \), 3.3.148.1, 3.3.1369.1, 6.0.4498621616.2, 6.0.384913312019.2, 9.9.6075640136512.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ R

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
2Data not computed
$37$37.6.4.1$x^{6} + 518 x^{3} + 171125$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
37.12.10.1$x^{12} + 1998 x^{6} + 21390625$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$59$59.6.3.2$x^{6} - 3481 x^{2} + 3491443$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
59.12.6.1$x^{12} + 9447434 x^{6} - 714924299 x^{2} + 22313502296089$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$