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Properties

Label 18.0.25874937625...2003.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 83^{9}$
Root discriminant $33.34$
Ramified primes $7, 83$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $D_9$ (as 18T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19900, -4870, -3111, 9905, 13457, -32340, 31261, -24727, 16214, -8957, 5274, -3691, 2498, -1323, 557, -188, 49, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 49*x^16 - 188*x^15 + 557*x^14 - 1323*x^13 + 2498*x^12 - 3691*x^11 + 5274*x^10 - 8957*x^9 + 16214*x^8 - 24727*x^7 + 31261*x^6 - 32340*x^5 + 13457*x^4 + 9905*x^3 - 3111*x^2 - 4870*x + 19900)
 
gp: K = bnfinit(x^18 - 9*x^17 + 49*x^16 - 188*x^15 + 557*x^14 - 1323*x^13 + 2498*x^12 - 3691*x^11 + 5274*x^10 - 8957*x^9 + 16214*x^8 - 24727*x^7 + 31261*x^6 - 32340*x^5 + 13457*x^4 + 9905*x^3 - 3111*x^2 - 4870*x + 19900, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 49 x^{16} - 188 x^{15} + 557 x^{14} - 1323 x^{13} + 2498 x^{12} - 3691 x^{11} + 5274 x^{10} - 8957 x^{9} + 16214 x^{8} - 24727 x^{7} + 31261 x^{6} - 32340 x^{5} + 13457 x^{4} + 9905 x^{3} - 3111 x^{2} - 4870 x + 19900 \)

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:  \(-2587493762586279810794282003=-\,7^{12}\cdot 83^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.34$
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:  $7, 83$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{5} - \frac{1}{16} a^{4} - \frac{7}{16} a^{3} - \frac{7}{16} a^{2} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{16} a^{11} - \frac{1}{8} a^{9} + \frac{1}{16} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{3}{16} a^{5} + \frac{1}{8} a^{3} + \frac{3}{16} a^{2} + \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{10} + \frac{1}{32} a^{8} + \frac{3}{16} a^{7} - \frac{5}{32} a^{6} + \frac{1}{8} a^{5} + \frac{1}{32} a^{4} - \frac{3}{8} a^{3} + \frac{7}{32} a^{2} + \frac{3}{16} a - \frac{1}{8}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{11} + \frac{1}{32} a^{9} - \frac{1}{16} a^{8} - \frac{5}{32} a^{7} + \frac{1}{8} a^{6} + \frac{1}{32} a^{5} + \frac{1}{8} a^{4} + \frac{7}{32} a^{3} + \frac{7}{16} a^{2} + \frac{3}{8} a$, $\frac{1}{1280} a^{14} - \frac{7}{1280} a^{13} + \frac{3}{640} a^{12} - \frac{5}{256} a^{11} + \frac{1}{640} a^{10} + \frac{119}{1280} a^{9} + \frac{1}{10} a^{8} + \frac{29}{256} a^{7} - \frac{143}{640} a^{6} - \frac{11}{256} a^{5} - \frac{51}{640} a^{4} + \frac{201}{1280} a^{3} - \frac{229}{1280} a^{2} + \frac{211}{640} a - \frac{11}{64}$, $\frac{1}{1280} a^{15} - \frac{3}{1280} a^{13} + \frac{17}{1280} a^{12} + \frac{27}{1280} a^{11} - \frac{27}{1280} a^{10} + \frac{41}{1280} a^{9} + \frac{81}{1280} a^{8} + \frac{49}{1280} a^{7} - \frac{297}{1280} a^{6} + \frac{273}{1280} a^{5} - \frac{193}{1280} a^{4} - \frac{71}{640} a^{3} + \frac{579}{1280} a^{2} - \frac{73}{640} a - \frac{29}{64}$, $\frac{1}{3579074368000} a^{16} - \frac{1}{447384296000} a^{15} + \frac{333116219}{894768592000} a^{14} - \frac{1165906749}{447384296000} a^{13} - \frac{386833427}{143162974720} a^{12} - \frac{22206416209}{1789537184000} a^{11} + \frac{7292392651}{715814873600} a^{10} + \frac{94459192517}{1789537184000} a^{9} + \frac{108843512833}{3579074368000} a^{8} + \frac{168444380483}{1789537184000} a^{7} - \frac{44546337773}{715814873600} a^{6} + \frac{297317865259}{1789537184000} a^{5} - \frac{17764028287}{447384296000} a^{4} - \frac{870214861303}{1789537184000} a^{3} - \frac{390876812909}{3579074368000} a^{2} - \frac{695912610357}{1789537184000} a + \frac{79183061723}{178953718400}$, $\frac{1}{27061381296448000} a^{17} + \frac{943}{6765345324112000} a^{16} - \frac{93004869657}{13530690648224000} a^{15} - \frac{1034598946753}{6765345324112000} a^{14} + \frac{56042946770903}{5412276259289600} a^{13} - \frac{105566251974717}{6765345324112000} a^{12} - \frac{135001454195587}{5412276259289600} a^{11} + \frac{95338976594123}{3382672662056000} a^{10} + \frac{101893841461703}{27061381296448000} a^{9} - \frac{673194216604661}{6765345324112000} a^{8} - \frac{1154232984183287}{5412276259289600} a^{7} - \frac{586451584070779}{3382672662056000} a^{6} - \frac{3157007559786053}{13530690648224000} a^{5} - \frac{800933571854817}{3382672662056000} a^{4} + \frac{8556231705105011}{27061381296448000} a^{3} + \frac{977144150704729}{6765345324112000} a^{2} - \frac{256247573723059}{676534532411200} a + \frac{58326958213599}{135306906482240}$

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:  $8$
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:  \( 5709912.4286 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9$ (as 18T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $D_9$
Character table for $D_9$

Intermediate fields

\(\Q(\sqrt{-83}) \), 3.1.83.1 x3, 6.0.571787.1, 9.1.5583424007329.1 x9

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

Sibling fields

Degree 9 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$7$7.9.6.3$x^{9} - 14 x^{6} + 49 x^{3} - 1372$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
7.9.6.3$x^{9} - 14 x^{6} + 49 x^{3} - 1372$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
$83$83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$