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Properties

Label 18.12.4880105390...9375.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,5^{9}\cdot 7^{12}\cdot 41^{2}\cdot 251\cdot 65409469^{2}$
Root discriminant $124.11$
Ramified primes $5, 7, 41, 251, 65409469$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 18T857

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![329341949, 2061278913, 4623947060, 5616772596, 3892288402, 1244201712, -152131512, -252477790, -52777043, 13144625, 6132831, -72891, -302453, -15681, 8312, 548, -132, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 132*x^16 + 548*x^15 + 8312*x^14 - 15681*x^13 - 302453*x^12 - 72891*x^11 + 6132831*x^10 + 13144625*x^9 - 52777043*x^8 - 252477790*x^7 - 152131512*x^6 + 1244201712*x^5 + 3892288402*x^4 + 5616772596*x^3 + 4623947060*x^2 + 2061278913*x + 329341949)
 
gp: K = bnfinit(x^18 - 6*x^17 - 132*x^16 + 548*x^15 + 8312*x^14 - 15681*x^13 - 302453*x^12 - 72891*x^11 + 6132831*x^10 + 13144625*x^9 - 52777043*x^8 - 252477790*x^7 - 152131512*x^6 + 1244201712*x^5 + 3892288402*x^4 + 5616772596*x^3 + 4623947060*x^2 + 2061278913*x + 329341949, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 132 x^{16} + 548 x^{15} + 8312 x^{14} - 15681 x^{13} - 302453 x^{12} - 72891 x^{11} + 6132831 x^{10} + 13144625 x^{9} - 52777043 x^{8} - 252477790 x^{7} - 152131512 x^{6} + 1244201712 x^{5} + 3892288402 x^{4} + 5616772596 x^{3} + 4623947060 x^{2} + 2061278913 x + 329341949 \)

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:  \(-48801053907192911222729739691193359375=-\,5^{9}\cdot 7^{12}\cdot 41^{2}\cdot 251\cdot 65409469^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $124.11$
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:  $5, 7, 41, 251, 65409469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{41} a^{15} + \frac{8}{41} a^{14} - \frac{14}{41} a^{13} - \frac{4}{41} a^{12} + \frac{20}{41} a^{11} - \frac{11}{41} a^{10} - \frac{13}{41} a^{9} + \frac{2}{41} a^{8} + \frac{17}{41} a^{7} - \frac{8}{41} a^{6} + \frac{4}{41} a^{5} - \frac{8}{41} a^{4} - \frac{15}{41} a^{2} + \frac{7}{41} a - \frac{10}{41}$, $\frac{1}{41} a^{16} + \frac{4}{41} a^{14} - \frac{15}{41} a^{13} + \frac{11}{41} a^{12} - \frac{7}{41} a^{11} - \frac{7}{41} a^{10} - \frac{17}{41} a^{9} + \frac{1}{41} a^{8} + \frac{20}{41} a^{7} - \frac{14}{41} a^{6} + \frac{1}{41} a^{5} - \frac{18}{41} a^{4} - \frac{15}{41} a^{3} + \frac{4}{41} a^{2} + \frac{16}{41} a - \frac{2}{41}$, $\frac{1}{773556814131221769750562773183300090688071656729413} a^{17} - \frac{8627141456087608574552391375744193554098664833491}{773556814131221769750562773183300090688071656729413} a^{16} + \frac{9407082054163486838752797562359123359758486758276}{773556814131221769750562773183300090688071656729413} a^{15} + \frac{166536557250950230518438671618601413464733185325051}{773556814131221769750562773183300090688071656729413} a^{14} + \frac{24220280381502052162253017883976969471445598604863}{773556814131221769750562773183300090688071656729413} a^{13} - \frac{17674794196034465134966513435895525671540366474682}{773556814131221769750562773183300090688071656729413} a^{12} + \frac{6828108409930965962686292433279636071969695097084}{18867239369054189506111287150812197333855406261693} a^{11} + \frac{162166565092193558245566448356407270102717560896972}{773556814131221769750562773183300090688071656729413} a^{10} + \frac{186400108274165331981480447276825426823948855492421}{773556814131221769750562773183300090688071656729413} a^{9} + \frac{323054029566668221162311958762061692570770825516489}{773556814131221769750562773183300090688071656729413} a^{8} - \frac{181970188041988511098711490171315401043776252541621}{773556814131221769750562773183300090688071656729413} a^{7} + \frac{119668411092086636888403808781863642553413621719133}{773556814131221769750562773183300090688071656729413} a^{6} - \frac{262102164377454241514206741736159684837456237605357}{773556814131221769750562773183300090688071656729413} a^{5} + \frac{214127957916014167378290991365845427455251879851631}{773556814131221769750562773183300090688071656729413} a^{4} + \frac{324074987558868833391369010482636758162390967692601}{773556814131221769750562773183300090688071656729413} a^{3} + \frac{62847756787136413974866022302510343037628385622803}{773556814131221769750562773183300090688071656729413} a^{2} + \frac{294090156420707500744314939342269467266343185329720}{773556814131221769750562773183300090688071656729413} a + \frac{67875451595566220941000697139515408314817246854108}{773556814131221769750562773183300090688071656729413}$

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:  \( 1102666415190 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T857:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 279936
The 159 conjugacy class representatives for t18n857 are not computed
Character table for t18n857 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 6.6.300125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ $18$ R R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $18$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ R ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.3.2.1$x^{3} - 41$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
251Data not computed
65409469Data not computed