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Properties

Label 18.0.16877848680...4907.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 19^{12}$
Root discriminant $37.00$
Ramified primes $3, 19$
Class number $27$ (GRH)
Class group $[3, 3, 3]$ (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![117649, 0, 0, 66885, 0, 0, 37339, 0, 0, 1076, 0, 0, 199, 0, 0, -2, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^15 + 199*x^12 + 1076*x^9 + 37339*x^6 + 66885*x^3 + 117649)
 
gp: K = bnfinit(x^18 - 2*x^15 + 199*x^12 + 1076*x^9 + 37339*x^6 + 66885*x^3 + 117649, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{15} + 199 x^{12} + 1076 x^{9} + 37339 x^{6} + 66885 x^{3} + 117649 \)

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:  \(-16877848680315122776257224907=-\,3^{27}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.00$
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:  $3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(171=3^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{171}(64,·)$, $\chi_{171}(1,·)$, $\chi_{171}(68,·)$, $\chi_{171}(134,·)$, $\chi_{171}(7,·)$, $\chi_{171}(11,·)$, $\chi_{171}(140,·)$, $\chi_{171}(77,·)$, $\chi_{171}(83,·)$, $\chi_{171}(20,·)$, $\chi_{171}(26,·)$, $\chi_{171}(163,·)$, $\chi_{171}(106,·)$, $\chi_{171}(49,·)$, $\chi_{171}(115,·)$, $\chi_{171}(121,·)$, $\chi_{171}(58,·)$, $\chi_{171}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a^{4} + \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{5} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{6} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} + \frac{1}{7} a$, $\frac{1}{7} a^{11} + \frac{1}{7} a^{2}$, $\frac{1}{259} a^{12} + \frac{4}{259} a^{9} - \frac{109}{259} a^{6} + \frac{19}{259} a^{3} - \frac{9}{37}$, $\frac{1}{259} a^{13} + \frac{4}{259} a^{10} + \frac{2}{259} a^{7} - \frac{92}{259} a^{4} + \frac{48}{259} a$, $\frac{1}{1813} a^{14} - \frac{107}{1813} a^{11} - \frac{109}{1813} a^{8} + \frac{796}{1813} a^{5} + \frac{862}{1813} a^{2}$, $\frac{1}{95997655621} a^{15} - \frac{124809911}{95997655621} a^{12} - \frac{3333924913}{95997655621} a^{9} - \frac{25578465796}{95997655621} a^{6} - \frac{25881049809}{95997655621} a^{3} - \frac{58613089}{279876547}$, $\frac{1}{671983589347} a^{16} + \frac{987132046}{671983589347} a^{13} - \frac{12600107888}{671983589347} a^{10} - \frac{37068532685}{671983589347} a^{7} - \frac{18468103429}{671983589347} a^{4} + \frac{17029221}{1959135829} a$, $\frac{1}{4703885125429} a^{17} + \frac{987132046}{4703885125429} a^{14} - \frac{12600107888}{4703885125429} a^{11} + \frac{154926778557}{4703885125429} a^{8} - \frac{210463414671}{4703885125429} a^{5} - \frac{5300625172}{13713950803} a^{2}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}$, which has order $27$ (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:  \( \frac{2018}{13713950803} a^{17} - \frac{200791}{13713950803} a^{14} + \frac{1068127}{13713950803} a^{11} - \frac{37675051}{13713950803} a^{8} - \frac{1377285}{279876547} a^{5} - \frac{9317434587}{13713950803} a^{2} \) (order $18$)
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:  \( 1472619.0824 \) (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

\(\Q(\sqrt{-3}) \), 3.3.29241.2, 3.3.29241.1, 3.3.361.1, \(\Q(\zeta_{9})^+\), 6.0.2565108243.2, 6.0.2565108243.1, 6.0.3518667.1, \(\Q(\zeta_{9})\), 9.9.25002110044521.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$19$19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$