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Properties

Label 18.6.11885973350...2736.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 3^{24}\cdot 7^{13}\cdot 13^{9}$
Root discriminant $100.96$
Ramified primes $2, 3, 7, 13$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T585

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5824, 0, -13104, 0, 501228, 0, 429843, 0, 36774, 0, -42246, 0, -8463, 0, 90, 0, 45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 45*x^16 + 90*x^14 - 8463*x^12 - 42246*x^10 + 36774*x^8 + 429843*x^6 + 501228*x^4 - 13104*x^2 - 5824)
 
gp: K = bnfinit(x^18 + 45*x^16 + 90*x^14 - 8463*x^12 - 42246*x^10 + 36774*x^8 + 429843*x^6 + 501228*x^4 - 13104*x^2 - 5824, 1)
 

Normalized defining polynomial

\( x^{18} + 45 x^{16} + 90 x^{14} - 8463 x^{12} - 42246 x^{10} + 36774 x^{8} + 429843 x^{6} + 501228 x^{4} - 13104 x^{2} - 5824 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1188597335032006811881326950728052736=2^{12}\cdot 3^{24}\cdot 7^{13}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $100.96$
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:  $2, 3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{6} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{396} a^{12} + \frac{5}{66} a^{10} + \frac{1}{33} a^{8} - \frac{1}{36} a^{6} + \frac{65}{132} a^{4} - \frac{61}{132} a^{2} - \frac{20}{99}$, $\frac{1}{396} a^{13} + \frac{5}{66} a^{11} + \frac{1}{33} a^{9} - \frac{1}{36} a^{7} + \frac{65}{132} a^{5} - \frac{61}{132} a^{3} - \frac{20}{99} a$, $\frac{1}{792} a^{14} - \frac{1}{792} a^{12} + \frac{1}{132} a^{10} + \frac{13}{792} a^{8} + \frac{1}{99} a^{6} - \frac{1}{2} a^{5} - \frac{1}{33} a^{4} + \frac{49}{792} a^{2} - \frac{1}{2} a + \frac{59}{198}$, $\frac{1}{1584} a^{15} + \frac{1}{1584} a^{13} + \frac{1}{24} a^{11} - \frac{95}{1584} a^{9} + \frac{59}{792} a^{7} - \frac{5}{264} a^{5} - \frac{1}{2} a^{4} + \frac{607}{1584} a^{3} - \frac{1}{2} a^{2} - \frac{113}{396} a$, $\frac{1}{2531485641470968512} a^{16} - \frac{954209596062823}{2531485641470968512} a^{14} - \frac{810489001004797}{1265742820735484256} a^{12} - \frac{3999441246879157}{230135058315542592} a^{10} - \frac{66193728643721693}{1265742820735484256} a^{8} + \frac{38707608814812695}{1265742820735484256} a^{6} - \frac{894733933944447077}{2531485641470968512} a^{4} - \frac{1}{2} a^{3} - \frac{41163598494827443}{158217852591935532} a^{2} - \frac{1}{2} a - \frac{45233558466302279}{158217852591935532}$, $\frac{1}{5062971282941937024} a^{17} - \frac{954209596062823}{5062971282941937024} a^{15} - \frac{810489001004797}{2531485641470968512} a^{13} - \frac{3999441246879157}{460270116631085184} a^{11} - \frac{66193728643721693}{2531485641470968512} a^{9} - \frac{172249527974434681}{2531485641470968512} a^{7} - \frac{894733933944447077}{5062971282941937024} a^{5} - \frac{120272524790795209}{316435705183871064} a^{3} - \frac{97972842663614123}{316435705183871064} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 175959345551 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T585:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 13824
The 96 conjugacy class representatives for t18n585 are not computed
Character table for t18n585 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.1785733746591249.2

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

Sibling fields

Degree 18 siblings: 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.6.4$x^{6} + x^{2} + 1$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
$3$3.9.12.4$x^{9} + 6 x^{6} + 18 x^{5} + 36 x^{3} + 27$$3$$3$$12$$C_3^2:C_3$$[2, 2]^{3}$
3.9.12.4$x^{9} + 6 x^{6} + 18 x^{5} + 36 x^{3} + 27$$3$$3$$12$$C_3^2:C_3$$[2, 2]^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$