# Properties

 Label 18.0.548...939.1 Degree $18$ Signature $[0, 9]$ Discriminant $-5.480\times 10^{21}$ Root discriminant $16.13$ Ramified prime $19$ Class number $1$ Class group trivial Galois group $C_{18}$ (as 18T1)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)

gp: K = bnfinit(x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]);

$$x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $18$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 9]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-5480386857784802185939$$$$\medspace = -\,19^{17}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $16.13$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $19$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $18$ This field is Galois and abelian over $\Q$. Conductor: $$19$$ Dirichlet character group: $\lbrace$$\chi_{19}(1,·), \chi_{19}(2,·), \chi_{19}(3,·), \chi_{19}(4,·), \chi_{19}(5,·), \chi_{19}(6,·), \chi_{19}(7,·), \chi_{19}(8,·), \chi_{19}(9,·), \chi_{19}(10,·), \chi_{19}(11,·), \chi_{19}(12,·), \chi_{19}(13,·), \chi_{19}(14,·), \chi_{19}(15,·), \chi_{19}(16,·), \chi_{19}(17,·)$$\chi_{19}(18,·)$$\rbrace$ This is a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $8$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$a$$ (order $38$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$a^{12} - a^{11}$$,  $$a^{12} + a^{2}$$,  $$a^{7} - a^{2}$$,  $$a^{16} - a^{9} + a^{4}$$,  $$a^{13} - a^{12} + a^{11} - a^{10} + a^{9} - a^{8} + a^{7} - a^{6} + a^{5} - a^{4} + a^{3} - a^{2} + a$$,  $$a^{9} - a^{8} + a^{7}$$,  $$a^{17} - a^{16} + a^{15} - a^{14} - a^{12} + a^{11} + a^{9} - a^{8} + a^{7} - a^{6} - a^{4} + a^{3} + a - 1$$,  $$a^{17} - 1$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$22305.8950792$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{9}\cdot 22305.8950792 \cdot 1}{38\sqrt{5480386857784802185939}}\approx 0.121017803346$

## Galois group

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

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A cyclic group of order 18 The 18 conjugacy class representatives for $C_{18}$ Character table for $C_{18}$

## Intermediate fields

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 $18$ $18$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ $18$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])

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];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
19Data not computed