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Properties

Label 18.18.1904329526...9837.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{27}\cdot 7^{12}\cdot 71^{5}$
Root discriminant $62.13$
Ramified primes $3, 7, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times A_4^2$ (as 18T109)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1081576, -3027996, -428712, 5597467, -1103103, -4350009, 1168166, 1853487, -501531, -471036, 116451, 72450, -15735, -6516, 1245, 311, -54, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 54*x^16 + 311*x^15 + 1245*x^14 - 6516*x^13 - 15735*x^12 + 72450*x^11 + 116451*x^10 - 471036*x^9 - 501531*x^8 + 1853487*x^7 + 1168166*x^6 - 4350009*x^5 - 1103103*x^4 + 5597467*x^3 - 428712*x^2 - 3027996*x + 1081576)
 
gp: K = bnfinit(x^18 - 6*x^17 - 54*x^16 + 311*x^15 + 1245*x^14 - 6516*x^13 - 15735*x^12 + 72450*x^11 + 116451*x^10 - 471036*x^9 - 501531*x^8 + 1853487*x^7 + 1168166*x^6 - 4350009*x^5 - 1103103*x^4 + 5597467*x^3 - 428712*x^2 - 3027996*x + 1081576, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 54 x^{16} + 311 x^{15} + 1245 x^{14} - 6516 x^{13} - 15735 x^{12} + 72450 x^{11} + 116451 x^{10} - 471036 x^{9} - 501531 x^{8} + 1853487 x^{7} + 1168166 x^{6} - 4350009 x^{5} - 1103103 x^{4} + 5597467 x^{3} - 428712 x^{2} - 3027996 x + 1081576 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(190432952662359521950134031359837=3^{27}\cdot 7^{12}\cdot 71^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.13$
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:  $3, 7, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{284} a^{16} - \frac{13}{284} a^{15} + \frac{7}{284} a^{14} + \frac{21}{71} a^{13} - \frac{121}{284} a^{12} + \frac{47}{284} a^{11} + \frac{31}{142} a^{10} - \frac{55}{142} a^{9} + \frac{57}{284} a^{8} - \frac{103}{284} a^{7} - \frac{31}{71} a^{6} + \frac{85}{284} a^{5} + \frac{75}{284} a^{4} + \frac{17}{71} a^{3} + \frac{67}{284} a^{2} - \frac{63}{142} a + \frac{34}{71}$, $\frac{1}{7385873404078430477142132958688444} a^{17} + \frac{9061902935136097813385742241467}{7385873404078430477142132958688444} a^{16} - \frac{1518086304652771305213798403404547}{7385873404078430477142132958688444} a^{15} - \frac{90753108416676582975880300852467}{3692936702039215238571066479344222} a^{14} - \frac{2710565160389119178067571990547367}{7385873404078430477142132958688444} a^{13} - \frac{717118596703705929235263461958825}{7385873404078430477142132958688444} a^{12} + \frac{339540201263078565467755294430162}{1846468351019607619285533239672111} a^{11} - \frac{408100064820577794903752798176148}{1846468351019607619285533239672111} a^{10} + \frac{2126974888020967121767043243945969}{7385873404078430477142132958688444} a^{9} + \frac{3436519203314621768841160796717661}{7385873404078430477142132958688444} a^{8} + \frac{196805905841126229946431346938103}{3692936702039215238571066479344222} a^{7} + \frac{280444647066203553555134249704471}{7385873404078430477142132958688444} a^{6} - \frac{1097581733182071231611631990282225}{7385873404078430477142132958688444} a^{5} + \frac{1358564422380389092874876526606713}{3692936702039215238571066479344222} a^{4} - \frac{1155014110724694742838678884808863}{7385873404078430477142132958688444} a^{3} - \frac{1366030949089563590668622131351277}{3692936702039215238571066479344222} a^{2} + \frac{1347419059544616190400948688412523}{3692936702039215238571066479344222} a - \frac{890431427677992617121430946022540}{1846468351019607619285533239672111}$

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:  $17$
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:  \( 26223924303.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times A_4^2$ (as 18T109):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 288
The 32 conjugacy class representatives for $C_2\times A_4^2$
Character table for $C_2\times A_4^2$ is not computed

Intermediate fields

3.3.3969.2, 3.3.3969.1, \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 9.9.62523502209.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
3Data not computed
7Data not computed
71Data not computed