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Properties

Label 18.0.25435288916...8867.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 11^{9}\cdot 19^{17}$
Root discriminant $92.68$
Ramified primes $3, 11, 19$
Class number $411328$ (GRH)
Class group $[2, 2, 4, 25708]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10715392153, -8165255321, 8165255321, -3383748761, 3383748761, -753920153, 753920153, -96463001, 96463001, -7432345, 7432345, -350361, 350361, -9881, 9881, -153, 153, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 153*x^16 - 153*x^15 + 9881*x^14 - 9881*x^13 + 350361*x^12 - 350361*x^11 + 7432345*x^10 - 7432345*x^9 + 96463001*x^8 - 96463001*x^7 + 753920153*x^6 - 753920153*x^5 + 3383748761*x^4 - 3383748761*x^3 + 8165255321*x^2 - 8165255321*x + 10715392153)
 
gp: K = bnfinit(x^18 - x^17 + 153*x^16 - 153*x^15 + 9881*x^14 - 9881*x^13 + 350361*x^12 - 350361*x^11 + 7432345*x^10 - 7432345*x^9 + 96463001*x^8 - 96463001*x^7 + 753920153*x^6 - 753920153*x^5 + 3383748761*x^4 - 3383748761*x^3 + 8165255321*x^2 - 8165255321*x + 10715392153, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 153 x^{16} - 153 x^{15} + 9881 x^{14} - 9881 x^{13} + 350361 x^{12} - 350361 x^{11} + 7432345 x^{10} - 7432345 x^{9} + 96463001 x^{8} - 96463001 x^{7} + 753920153 x^{6} - 753920153 x^{5} + 3383748761 x^{4} - 3383748761 x^{3} + 8165255321 x^{2} - 8165255321 x + 10715392153 \)

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:  \(-254352889166747560743047516520738867=-\,3^{9}\cdot 11^{9}\cdot 19^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.68$
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:  $3, 11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(627=3\cdot 11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{627}(1,·)$, $\chi_{627}(65,·)$, $\chi_{627}(395,·)$, $\chi_{627}(397,·)$, $\chi_{627}(527,·)$, $\chi_{627}(529,·)$, $\chi_{627}(595,·)$, $\chi_{627}(100,·)$, $\chi_{627}(463,·)$, $\chi_{627}(32,·)$, $\chi_{627}(98,·)$, $\chi_{627}(164,·)$, $\chi_{627}(230,·)$, $\chi_{627}(232,·)$, $\chi_{627}(199,·)$, $\chi_{627}(428,·)$, $\chi_{627}(626,·)$, $\chi_{627}(562,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{1869986953} a^{10} + \frac{670841598}{1869986953} a^{9} + \frac{80}{1869986953} a^{8} - \frac{319065722}{1869986953} a^{7} + \frac{2240}{1869986953} a^{6} - \frac{177629516}{1869986953} a^{5} + \frac{25600}{1869986953} a^{4} + \frac{498834250}{1869986953} a^{3} + \frac{102400}{1869986953} a^{2} - \frac{672784753}{1869986953} a + \frac{65536}{1869986953}$, $\frac{1}{1869986953} a^{11} + \frac{88}{1869986953} a^{9} + \frac{243228075}{1869986953} a^{8} + \frac{2816}{1869986953} a^{7} + \frac{606701176}{1869986953} a^{6} + \frac{39424}{1869986953} a^{5} + \frac{914101802}{1869986953} a^{4} + \frac{225280}{1869986953} a^{3} - \frac{881701498}{1869986953} a^{2} + \frac{360448}{1869986953} a - \frac{881701498}{1869986953}$, $\frac{1}{1869986953} a^{12} - \frac{821237006}{1869986953} a^{9} - \frac{4224}{1869986953} a^{8} + \frac{634680417}{1869986953} a^{7} - \frac{157696}{1869986953} a^{6} - \frac{284383367}{1869986953} a^{5} - \frac{2027520}{1869986953} a^{4} + \frac{100571374}{1869986953} a^{3} - \frac{8650752}{1869986953} a^{2} + \frac{353761223}{1869986953} a - \frac{5767168}{1869986953}$, $\frac{1}{1869986953} a^{13} - \frac{4992}{1869986953} a^{9} + \frac{884097542}{1869986953} a^{8} - \frac{212992}{1869986953} a^{7} - \frac{780651679}{1869986953} a^{6} - \frac{3354624}{1869986953} a^{5} - \frac{495387605}{1869986953} a^{4} - \frac{20447232}{1869986953} a^{3} - \frac{160087740}{1869986953} a^{2} - \frac{34078720}{1869986953} a + \frac{493930923}{1869986953}$, $\frac{1}{1869986953} a^{14} + \frac{578721935}{1869986953} a^{9} + \frac{186368}{1869986953} a^{8} - \frac{327851947}{1869986953} a^{7} + \frac{7827456}{1869986953} a^{6} - \frac{848115755}{1869986953} a^{5} + \frac{107347968}{1869986953} a^{4} - \frac{802133136}{1869986953} a^{3} + \frac{477102080}{1869986953} a^{2} + \frac{449011535}{1869986953} a + \frac{327155712}{1869986953}$, $\frac{1}{1869986953} a^{15} + \frac{232960}{1869986953} a^{9} + \frac{124067078}{1869986953} a^{8} + \frac{11182080}{1869986953} a^{7} + \frac{585695227}{1869986953} a^{6} + \frac{187858944}{1869986953} a^{5} - \frac{177040517}{1869986953} a^{4} - \frac{677231753}{1869986953} a^{3} - \frac{790591895}{1869986953} a^{2} + \frac{174736247}{1869986953} a - \frac{45351414}{1869986953}$, $\frac{1}{1869986953} a^{16} - \frac{584966886}{1869986953} a^{9} - \frac{7454720}{1869986953} a^{8} + \frac{24897550}{1869986953} a^{7} - \frac{333971456}{1869986953} a^{6} - \frac{546276094}{1869986953} a^{5} + \frac{838940059}{1869986953} a^{4} - \frac{748264663}{1869986953} a^{3} + \frac{629462636}{1869986953} a^{2} + \frac{804228724}{1869986953} a - \frac{307370936}{1869986953}$, $\frac{1}{1869986953} a^{17} - \frac{9748480}{1869986953} a^{9} + \frac{72574605}{1869986953} a^{8} - \frac{499122176}{1869986953} a^{7} + \frac{788681446}{1869986953} a^{6} + \frac{615296685}{1869986953} a^{5} - \frac{451502687}{1869986953} a^{4} + \frac{927061143}{1869986953} a^{3} + \frac{121289675}{1869986953} a^{2} - \frac{715126691}{1869986953} a - \frac{212682557}{1869986953}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{25708}$, which has order $411328$ (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:  \( 22305.8950792 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

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

Intermediate fields

\(\Q(\sqrt{-627}) \), 3.3.361.1, 6.0.88983569763.2, \(\Q(\zeta_{19})^+\)

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 $18$ R $18$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R $18$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ $18$ $18$ $18$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
3Data not computed
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19Data not computed