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Properties

Label 18.0.82978860458...9791.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,11^{12}\cdot 31^{9}$
Root discriminant $27.54$
Ramified primes $11, 31$
Class number $1$ (GRH)
Class group Trivial (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![19127, 7750, -16172, -8374, 9952, 1065, -796, 2377, 3164, 918, 131, 47, 90, -203, 83, -48, 18, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 18*x^16 - 48*x^15 + 83*x^14 - 203*x^13 + 90*x^12 + 47*x^11 + 131*x^10 + 918*x^9 + 3164*x^8 + 2377*x^7 - 796*x^6 + 1065*x^5 + 9952*x^4 - 8374*x^3 - 16172*x^2 + 7750*x + 19127)
 
gp: K = bnfinit(x^18 - 3*x^17 + 18*x^16 - 48*x^15 + 83*x^14 - 203*x^13 + 90*x^12 + 47*x^11 + 131*x^10 + 918*x^9 + 3164*x^8 + 2377*x^7 - 796*x^6 + 1065*x^5 + 9952*x^4 - 8374*x^3 - 16172*x^2 + 7750*x + 19127, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 18 x^{16} - 48 x^{15} + 83 x^{14} - 203 x^{13} + 90 x^{12} + 47 x^{11} + 131 x^{10} + 918 x^{9} + 3164 x^{8} + 2377 x^{7} - 796 x^{6} + 1065 x^{5} + 9952 x^{4} - 8374 x^{3} - 16172 x^{2} + 7750 x + 19127 \)

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:  \(-82978860458831265178139791=-\,11^{12}\cdot 31^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.54$
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:  $11, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{13} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{2}{9} a^{6} - \frac{1}{9} a^{4} - \frac{2}{9} a^{3} - \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{549} a^{15} - \frac{52}{549} a^{13} - \frac{56}{549} a^{12} + \frac{26}{183} a^{11} - \frac{85}{549} a^{10} - \frac{5}{549} a^{9} - \frac{14}{183} a^{8} + \frac{4}{61} a^{7} - \frac{251}{549} a^{6} + \frac{80}{549} a^{5} - \frac{35}{183} a^{4} - \frac{2}{9} a^{3} - \frac{58}{549} a^{2} + \frac{11}{183} a - \frac{128}{549}$, $\frac{1}{18568827} a^{16} + \frac{326}{18568827} a^{15} + \frac{873529}{18568827} a^{14} - \frac{73619}{2063203} a^{13} + \frac{33512}{304407} a^{12} - \frac{1572857}{18568827} a^{11} + \frac{49220}{6189609} a^{10} + \frac{2891924}{18568827} a^{9} - \frac{1218223}{18568827} a^{8} + \frac{1028111}{18568827} a^{7} - \frac{312036}{2063203} a^{6} - \frac{7111757}{18568827} a^{5} - \frac{358201}{18568827} a^{4} - \frac{2442073}{6189609} a^{3} + \frac{8427673}{18568827} a^{2} + \frac{6328034}{18568827} a - \frac{8316866}{18568827}$, $\frac{1}{65173579766608936999245911793} a^{17} + \frac{1201305395844692490818}{65173579766608936999245911793} a^{16} + \frac{4590819428179235194661095}{65173579766608936999245911793} a^{15} + \frac{1965351051810763660270778444}{65173579766608936999245911793} a^{14} - \frac{6198951787891309633071160}{21724526588869645666415303931} a^{13} - \frac{3518709889458680723742386066}{65173579766608936999245911793} a^{12} - \frac{1814579882022335400500459200}{65173579766608936999245911793} a^{11} - \frac{1239460169409757879016376056}{21724526588869645666415303931} a^{10} + \frac{5527800930390248417966782421}{65173579766608936999245911793} a^{9} - \frac{6813779945558712090694675385}{65173579766608936999245911793} a^{8} - \frac{5562840499573568103190785368}{65173579766608936999245911793} a^{7} + \frac{9795102084437570580402136}{31900920101130169847893251} a^{6} + \frac{9916163025740139523242315188}{65173579766608936999245911793} a^{5} + \frac{30798510709027891329506230810}{65173579766608936999245911793} a^{4} - \frac{1236770424608036160194726284}{21724526588869645666415303931} a^{3} + \frac{16660075553591870538448590542}{65173579766608936999245911793} a^{2} - \frac{10973560022739841494138321013}{65173579766608936999245911793} a - \frac{10107369888725610125099719909}{65173579766608936999245911793}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 605404.210476 \) (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

\(\Q(\sqrt{-31}) \), 3.1.31.1 x3, 6.0.29791.1, 9.1.1636073786281.1 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$11$11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$