Properties

Label 18.0.81735093342...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{6}\cdot 5^{8}\cdot 83^{6}$
Root discriminant $11.24$
Ramified primes $2, 5, 83$
Class number $1$
Class group Trivial
Galois Group $C_2\times (C_3\times A_4):S_3$ (as 18T156)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -3, 0, 7, 0, 18, 0, 7, 0, 2, 0, -5, 0, -6, 0, 1, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + x^16 - 6*x^14 - 5*x^12 + 2*x^10 + 7*x^8 + 18*x^6 + 7*x^4 - 3*x^2 + 1)
gp: K = bnfinit(x^18 + x^16 - 6*x^14 - 5*x^12 + 2*x^10 + 7*x^8 + 18*x^6 + 7*x^4 - 3*x^2 + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut +\mathstrut x^{16} \) \(\mathstrut -\mathstrut 6 x^{14} \) \(\mathstrut -\mathstrut 5 x^{12} \) \(\mathstrut +\mathstrut 2 x^{10} \) \(\mathstrut +\mathstrut 7 x^{8} \) \(\mathstrut +\mathstrut 18 x^{6} \) \(\mathstrut +\mathstrut 7 x^{4} \) \(\mathstrut -\mathstrut 3 x^{2} \) \(\mathstrut +\mathstrut 1 \)

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:  \(-8173509334225000000=-\,2^{6}\cdot 5^{8}\cdot 83^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.24$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 5, 83$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{10} - \frac{1}{3} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{11} - \frac{1}{3} a^{9} - \frac{1}{2} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{6} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{6}$, $\frac{1}{6} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a$, $\frac{1}{582} a^{16} + \frac{7}{194} a^{14} + \frac{13}{291} a^{12} - \frac{1}{2} a^{11} + \frac{112}{291} a^{10} - \frac{1}{2} a^{9} + \frac{39}{194} a^{8} - \frac{136}{291} a^{6} - \frac{1}{2} a^{5} - \frac{281}{582} a^{4} - \frac{139}{291} a^{2} - \frac{17}{291}$, $\frac{1}{582} a^{17} + \frac{7}{194} a^{15} + \frac{13}{291} a^{13} + \frac{112}{291} a^{11} + \frac{39}{194} a^{9} + \frac{19}{582} a^{7} - \frac{1}{2} a^{6} + \frac{5}{291} a^{5} - \frac{1}{2} a^{4} - \frac{139}{291} a^{3} + \frac{257}{582} a - \frac{1}{2}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( \frac{81}{194} a^{17} - \frac{91}{582} a^{16} + \frac{253}{582} a^{15} - \frac{34}{291} a^{14} - \frac{721}{291} a^{13} + \frac{272}{291} a^{12} - \frac{1343}{582} a^{11} + \frac{277}{582} a^{10} + \frac{301}{582} a^{9} - \frac{57}{194} a^{8} + \frac{1901}{582} a^{7} - \frac{78}{97} a^{6} + \frac{2282}{291} a^{5} - \frac{1589}{582} a^{4} + \frac{2383}{582} a^{3} - \frac{19}{582} a^{2} - \frac{19}{97} a + \frac{63}{97} \),  \( \frac{49}{291} a^{17} - \frac{41}{194} a^{16} + \frac{59}{291} a^{15} - \frac{61}{582} a^{14} - \frac{278}{291} a^{13} + \frac{341}{291} a^{12} - \frac{649}{582} a^{11} + \frac{95}{291} a^{10} - \frac{77}{582} a^{9} - \frac{35}{582} a^{8} + \frac{446}{291} a^{7} - \frac{147}{97} a^{6} + \frac{747}{194} a^{5} - \frac{507}{194} a^{4} + \frac{180}{97} a^{3} + \frac{122}{291} a^{2} + \frac{80}{291} a + \frac{151}{291} \),  \( \frac{45}{194} a^{17} + \frac{11}{291} a^{16} + \frac{36}{97} a^{15} + \frac{57}{194} a^{14} - \frac{379}{291} a^{13} - \frac{5}{291} a^{12} - \frac{1091}{582} a^{11} - \frac{446}{291} a^{10} + \frac{89}{291} a^{9} - \frac{56}{97} a^{8} + \frac{458}{291} a^{7} + \frac{127}{582} a^{6} + \frac{1354}{291} a^{5} + \frac{1093}{582} a^{4} + \frac{926}{291} a^{3} + \frac{1016}{291} a^{2} - \frac{419}{582} a - \frac{83}{291} \),  \( \frac{55}{291} a^{17} - \frac{24}{97} a^{16} + \frac{88}{291} a^{15} - \frac{211}{582} a^{14} - \frac{535}{582} a^{13} + \frac{815}{582} a^{12} - \frac{871}{582} a^{11} + \frac{556}{291} a^{10} - \frac{419}{582} a^{9} - \frac{67}{582} a^{8} + \frac{635}{582} a^{7} - \frac{592}{291} a^{6} + \frac{2555}{582} a^{5} - \frac{1496}{291} a^{4} + \frac{703}{194} a^{3} - \frac{839}{291} a^{2} + \frac{273}{194} a + \frac{40}{97} \),  \( \frac{45}{194} a^{17} - \frac{11}{291} a^{16} + \frac{36}{97} a^{15} - \frac{57}{194} a^{14} - \frac{379}{291} a^{13} + \frac{5}{291} a^{12} - \frac{1091}{582} a^{11} + \frac{446}{291} a^{10} + \frac{89}{291} a^{9} + \frac{56}{97} a^{8} + \frac{458}{291} a^{7} - \frac{127}{582} a^{6} + \frac{1354}{291} a^{5} - \frac{1093}{582} a^{4} + \frac{926}{291} a^{3} - \frac{1016}{291} a^{2} - \frac{419}{582} a + \frac{83}{291} \),  \( \frac{91}{291} a^{17} + \frac{68}{291} a^{15} - \frac{544}{291} a^{13} - \frac{277}{291} a^{11} + \frac{57}{97} a^{9} + \frac{156}{97} a^{7} + \frac{1589}{291} a^{5} + \frac{310}{291} a^{3} - \frac{126}{97} a \),  \( \frac{43}{291} a^{17} + \frac{10}{97} a^{15} - \frac{80}{97} a^{13} - \frac{55}{97} a^{11} - \frac{110}{291} a^{9} + \frac{332}{291} a^{7} + \frac{305}{97} a^{5} + \frac{656}{291} a^{3} + \frac{127}{97} a \),  \( a \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 72.3431657313 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2\times (C_3\times A_4):S_3$ (as 18T156):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 432
The 38 conjugacy class representatives for $C_2\times (C_3\times A_4):S_3$
Character table for $C_2\times (C_3\times A_4):S_3$ is not computed

Intermediate fields

3.1.83.1, 6.0.27556.1, 9.3.357366875.1

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

Sibling fields

Degree 18 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.12.0.1$x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$5$5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.12.8.2$x^{12} + 25 x^{6} - 250 x^{3} + 1250$$3$$4$$8$$C_3\times (C_3 : C_4)$$[\ ]_{3}^{12}$
$83$83.6.0.1$x^{6} - x + 34$$1$$6$$0$$C_6$$[\ ]^{6}$
83.6.3.2$x^{6} - 6889 x^{2} + 1715361$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
83.6.3.2$x^{6} - 6889 x^{2} + 1715361$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$