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Properties

Label 18.6.11247392561...9689.4
Degree $18$
Signature $[6, 6]$
Discriminant $3^{24}\cdot 73^{5}\cdot 577^{3}$
Root discriminant $41.11$
Ramified primes $3, 73, 577$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T840

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![38699, -55662, 2343, 572016, 190836, 49869, -144749, -49698, -8640, 3507, 6294, -429, 1212, -552, 114, -76, 15, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 15*x^16 - 76*x^15 + 114*x^14 - 552*x^13 + 1212*x^12 - 429*x^11 + 6294*x^10 + 3507*x^9 - 8640*x^8 - 49698*x^7 - 144749*x^6 + 49869*x^5 + 190836*x^4 + 572016*x^3 + 2343*x^2 - 55662*x + 38699)
 
gp: K = bnfinit(x^18 - 3*x^17 + 15*x^16 - 76*x^15 + 114*x^14 - 552*x^13 + 1212*x^12 - 429*x^11 + 6294*x^10 + 3507*x^9 - 8640*x^8 - 49698*x^7 - 144749*x^6 + 49869*x^5 + 190836*x^4 + 572016*x^3 + 2343*x^2 - 55662*x + 38699, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 15 x^{16} - 76 x^{15} + 114 x^{14} - 552 x^{13} + 1212 x^{12} - 429 x^{11} + 6294 x^{10} + 3507 x^{9} - 8640 x^{8} - 49698 x^{7} - 144749 x^{6} + 49869 x^{5} + 190836 x^{4} + 572016 x^{3} + 2343 x^{2} - 55662 x + 38699 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(112473925614060022797462679689=3^{24}\cdot 73^{5}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.11$
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:  $3, 73, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{15} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{2}{9}$, $\frac{1}{9} a^{16} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{9} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{670179525072776021506757407407506361406164957} a^{17} + \frac{927595514040217063233871235364074342035802}{670179525072776021506757407407506361406164957} a^{16} + \frac{31980958897478350362869806515056702257303675}{670179525072776021506757407407506361406164957} a^{15} + \frac{11178171560034820939361330110484135283135742}{223393175024258673835585802469168787135388319} a^{14} - \frac{2849758884082553703540398977191974553143668}{74464391674752891278528600823056262378462773} a^{13} + \frac{35993073820465949723425865287917811271335891}{223393175024258673835585802469168787135388319} a^{12} + \frac{19025654195197915667103819187794120504866205}{74464391674752891278528600823056262378462773} a^{11} + \frac{29031749451391412833432305682996416397521133}{223393175024258673835585802469168787135388319} a^{10} + \frac{30593459413427011484649630620821335620560370}{74464391674752891278528600823056262378462773} a^{9} + \frac{51853644739072904411541788305629694948608620}{223393175024258673835585802469168787135388319} a^{8} - \frac{13277320350579103274289372819363458785740499}{74464391674752891278528600823056262378462773} a^{7} + \frac{14310283786090754489487698429418463346283992}{223393175024258673835585802469168787135388319} a^{6} - \frac{214383452870153191879139084579385258432107540}{670179525072776021506757407407506361406164957} a^{5} - \frac{156546170426944331213640190550048968266275470}{670179525072776021506757407407506361406164957} a^{4} - \frac{320466346992676153005301619964090707429220545}{670179525072776021506757407407506361406164957} a^{3} + \frac{64677685194503391317151314872589204951249206}{670179525072776021506757407407506361406164957} a^{2} + \frac{20036181812004733841202071889494278873158317}{670179525072776021506757407407506361406164957} a + \frac{205912007864486610597719246062925453161344175}{670179525072776021506757407407506361406164957}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 14980954.3626 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T840:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 165888
The 180 conjugacy class representatives for t18n840 are not computed
Character table for t18n840 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.22384826361.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
73Data not computed
577Data not computed