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Properties

Label 18.0.21706971846...2368.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 139^{10}$
Root discriminant $42.64$
Ramified primes $2, 3, 139$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1185076, -1634408, 675168, 494826, 142906, 137522, 60667, 59948, 16613, 7210, 580, 1100, 533, 70, 68, -18, 11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 11*x^16 - 18*x^15 + 68*x^14 + 70*x^13 + 533*x^12 + 1100*x^11 + 580*x^10 + 7210*x^9 + 16613*x^8 + 59948*x^7 + 60667*x^6 + 137522*x^5 + 142906*x^4 + 494826*x^3 + 675168*x^2 - 1634408*x + 1185076)
 
gp: K = bnfinit(x^18 + 11*x^16 - 18*x^15 + 68*x^14 + 70*x^13 + 533*x^12 + 1100*x^11 + 580*x^10 + 7210*x^9 + 16613*x^8 + 59948*x^7 + 60667*x^6 + 137522*x^5 + 142906*x^4 + 494826*x^3 + 675168*x^2 - 1634408*x + 1185076, 1)
 

Normalized defining polynomial

\( x^{18} + 11 x^{16} - 18 x^{15} + 68 x^{14} + 70 x^{13} + 533 x^{12} + 1100 x^{11} + 580 x^{10} + 7210 x^{9} + 16613 x^{8} + 59948 x^{7} + 60667 x^{6} + 137522 x^{5} + 142906 x^{4} + 494826 x^{3} + 675168 x^{2} - 1634408 x + 1185076 \)

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:  \(-217069718467413515899000762368=-\,2^{12}\cdot 3^{9}\cdot 139^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.64$
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:  $2, 3, 139$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{581504204017196514770555363948138361666864559012} a^{17} - \frac{12114927448103966622486267225754131350609069402}{145376051004299128692638840987034590416716139753} a^{16} + \frac{12806791227701387063333608664912679903165312545}{145376051004299128692638840987034590416716139753} a^{15} + \frac{18286637474112738550413732297854441745394441523}{581504204017196514770555363948138361666864559012} a^{14} - \frac{61729030415268463260004678194402869581279559677}{581504204017196514770555363948138361666864559012} a^{13} + \frac{7902027085931255944953192355726985216726527026}{145376051004299128692638840987034590416716139753} a^{12} + \frac{22782350551965031715770718381648091742264579468}{145376051004299128692638840987034590416716139753} a^{11} + \frac{4307653472719330234105783586737687214186332465}{581504204017196514770555363948138361666864559012} a^{10} - \frac{61614737650694036752774730976653707008633847149}{581504204017196514770555363948138361666864559012} a^{9} - \frac{26093084511006646703170138173640198445673727564}{145376051004299128692638840987034590416716139753} a^{8} + \frac{1816451956404383146557463769975809731240160947}{290752102008598257385277681974069180833432279506} a^{7} + \frac{40573815504329930341520392424911836193736174011}{581504204017196514770555363948138361666864559012} a^{6} - \frac{41165084385772212166948106257479894606346397127}{145376051004299128692638840987034590416716139753} a^{5} - \frac{19016887323022301754199908153293872889209431201}{290752102008598257385277681974069180833432279506} a^{4} + \frac{125013924598182517154135985253497434169542154139}{290752102008598257385277681974069180833432279506} a^{3} - \frac{31372005877488551336987564290420064076019724581}{145376051004299128692638840987034590416716139753} a^{2} - \frac{26069961191197688776687445836988743582200046237}{145376051004299128692638840987034590416716139753} a + \frac{30712443121840444366831930792867992833320513096}{145376051004299128692638840987034590416716139753}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( -\frac{89098187975460870611104}{107957197730529933785371458301} a^{17} + \frac{107354931511173656679081}{215914395461059867570742916602} a^{16} - \frac{1617000367406799472857149}{215914395461059867570742916602} a^{15} + \frac{1939101879461298428870325}{215914395461059867570742916602} a^{14} - \frac{8893978148857156263341025}{215914395461059867570742916602} a^{13} - \frac{13810012277400510776668239}{107957197730529933785371458301} a^{12} - \frac{13244465693408543263017709}{107957197730529933785371458301} a^{11} - \frac{240179175628784724910979467}{215914395461059867570742916602} a^{10} - \frac{35919571221702292942087357}{215914395461059867570742916602} a^{9} - \frac{773216720024351424713533754}{107957197730529933785371458301} a^{8} - \frac{1209424028294137411012618987}{107957197730529933785371458301} a^{7} - \frac{8904840087239695559033884499}{215914395461059867570742916602} a^{6} - \frac{10526492263261481024715218185}{215914395461059867570742916602} a^{5} - \frac{30006825664018924686827042731}{215914395461059867570742916602} a^{4} - \frac{48041395390349649234031799145}{215914395461059867570742916602} a^{3} - \frac{15230617268125957204719029568}{107957197730529933785371458301} a^{2} - \frac{133825957177886006278538289386}{107957197730529933785371458301} a + \frac{139260876226920615793108233060}{107957197730529933785371458301} \) (order $6$)
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:  \( 9350988.404859575 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.139.1, 6.0.8346672.1, 6.0.521667.1

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

Sibling fields

Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.2$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}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$139$139.3.2.2$x^{3} + 556$$3$$1$$2$$C_3$$[\ ]_{3}$
139.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
139.6.3.2$x^{6} - 19321 x^{2} + 13428095$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
139.6.5.4$x^{6} + 556$$6$$1$$5$$C_6$$[\ ]_{6}$