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Properties

Label 18.18.5983423332...3125.1
Degree $18$
Signature $[18, 0]$
Discriminant $5^{9}\cdot 7^{12}\cdot 19^{12}$
Root discriminant $58.26$
Ramified primes $5, 7, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21211, -34901, -169876, 264347, 400750, -612584, -267164, 534308, 33211, -215160, 22847, 43035, -8360, -4277, 1057, 195, -56, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 56*x^16 + 195*x^15 + 1057*x^14 - 4277*x^13 - 8360*x^12 + 43035*x^11 + 22847*x^10 - 215160*x^9 + 33211*x^8 + 534308*x^7 - 267164*x^6 - 612584*x^5 + 400750*x^4 + 264347*x^3 - 169876*x^2 - 34901*x + 21211)
 
gp: K = bnfinit(x^18 - 3*x^17 - 56*x^16 + 195*x^15 + 1057*x^14 - 4277*x^13 - 8360*x^12 + 43035*x^11 + 22847*x^10 - 215160*x^9 + 33211*x^8 + 534308*x^7 - 267164*x^6 - 612584*x^5 + 400750*x^4 + 264347*x^3 - 169876*x^2 - 34901*x + 21211, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 56 x^{16} + 195 x^{15} + 1057 x^{14} - 4277 x^{13} - 8360 x^{12} + 43035 x^{11} + 22847 x^{10} - 215160 x^{9} + 33211 x^{8} + 534308 x^{7} - 267164 x^{6} - 612584 x^{5} + 400750 x^{4} + 264347 x^{3} - 169876 x^{2} - 34901 x + 21211 \)

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:  \(59834233322368760002940158203125=5^{9}\cdot 7^{12}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.26$
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:  $5, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(665=5\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{665}(64,·)$, $\chi_{665}(1,·)$, $\chi_{665}(324,·)$, $\chi_{665}(134,·)$, $\chi_{665}(11,·)$, $\chi_{665}(144,·)$, $\chi_{665}(596,·)$, $\chi_{665}(39,·)$, $\chi_{665}(296,·)$, $\chi_{665}(106,·)$, $\chi_{665}(429,·)$, $\chi_{665}(239,·)$, $\chi_{665}(501,·)$, $\chi_{665}(121,·)$, $\chi_{665}(634,·)$, $\chi_{665}(571,·)$, $\chi_{665}(254,·)$, $\chi_{665}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{748931898040532589943310005571469749} a^{17} + \frac{233087758189870167316483781543566147}{748931898040532589943310005571469749} a^{16} - \frac{103443158809993198415946600167604534}{748931898040532589943310005571469749} a^{15} + \frac{147666749702851401387606920147382427}{748931898040532589943310005571469749} a^{14} - \frac{30847401937341419675933094276601518}{748931898040532589943310005571469749} a^{13} + \frac{279703524954546436329513326143164088}{748931898040532589943310005571469749} a^{12} - \frac{41707906391575413871861373695226823}{748931898040532589943310005571469749} a^{11} - \frac{144782849640098234428635641665311449}{748931898040532589943310005571469749} a^{10} - \frac{185877841282520327257200203632305953}{748931898040532589943310005571469749} a^{9} - \frac{81308884286696341678705157186624198}{748931898040532589943310005571469749} a^{8} + \frac{27713860378431924492342029822945627}{748931898040532589943310005571469749} a^{7} - \frac{329407069986158224281708046412215029}{748931898040532589943310005571469749} a^{6} - \frac{19831589632421791036114847268557954}{748931898040532589943310005571469749} a^{5} + \frac{314774261733195596457584447776775565}{748931898040532589943310005571469749} a^{4} + \frac{20452652778550350517092628379292721}{748931898040532589943310005571469749} a^{3} - \frac{14974597296183812121403574345446362}{748931898040532589943310005571469749} a^{2} + \frac{151345287399713469097034297692785481}{748931898040532589943310005571469749} a + \frac{162382408025814576795308535981173679}{748931898040532589943310005571469749}$

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:  \( 7678869702.19 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.361.1, 3.3.17689.2, \(\Q(\zeta_{7})^+\), 3.3.17689.1, 6.6.16290125.1, 6.6.39112590125.1, 6.6.300125.1, 6.6.39112590125.2, 9.9.5534900853769.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$19$19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$