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Properties

Label 18.0.67747954313...3632.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 7^{12}\cdot 37^{6}\cdot 167^{3}$
Root discriminant $45.42$
Ramified primes $2, 7, 37, 167$
Class number $216$ (GRH)
Class group $[216]$ (GRH)
Galois group $C_2\times S_3\times A_4$ (as 18T60)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![61027, -11895, 31305, -626, 29602, 895, 13566, -2817, 2987, -405, 1232, -92, -186, -15, 84, -5, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 10*x^16 - 5*x^15 + 84*x^14 - 15*x^13 - 186*x^12 - 92*x^11 + 1232*x^10 - 405*x^9 + 2987*x^8 - 2817*x^7 + 13566*x^6 + 895*x^5 + 29602*x^4 - 626*x^3 + 31305*x^2 - 11895*x + 61027)
 
gp: K = bnfinit(x^18 - 10*x^16 - 5*x^15 + 84*x^14 - 15*x^13 - 186*x^12 - 92*x^11 + 1232*x^10 - 405*x^9 + 2987*x^8 - 2817*x^7 + 13566*x^6 + 895*x^5 + 29602*x^4 - 626*x^3 + 31305*x^2 - 11895*x + 61027, 1)
 

Normalized defining polynomial

\( x^{18} - 10 x^{16} - 5 x^{15} + 84 x^{14} - 15 x^{13} - 186 x^{12} - 92 x^{11} + 1232 x^{10} - 405 x^{9} + 2987 x^{8} - 2817 x^{7} + 13566 x^{6} + 895 x^{5} + 29602 x^{4} - 626 x^{3} + 31305 x^{2} - 11895 x + 61027 \)

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:  \(-677479543132411644552981573632=-\,2^{12}\cdot 7^{12}\cdot 37^{6}\cdot 167^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.42$
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:  $2, 7, 37, 167$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{4} a^{11} - \frac{1}{2} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{24} a^{15} + \frac{1}{24} a^{14} + \frac{1}{12} a^{13} - \frac{5}{24} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{24} a^{9} - \frac{5}{24} a^{8} + \frac{3}{8} a^{6} - \frac{11}{24} a^{5} + \frac{7}{24} a^{4} - \frac{3}{8} a^{3} + \frac{1}{12} a^{2} - \frac{5}{12} a + \frac{7}{24}$, $\frac{1}{696} a^{16} - \frac{5}{696} a^{15} + \frac{37}{348} a^{14} + \frac{55}{696} a^{13} - \frac{73}{348} a^{12} - \frac{79}{348} a^{11} - \frac{151}{696} a^{10} + \frac{331}{696} a^{9} + \frac{1}{116} a^{8} + \frac{33}{232} a^{7} - \frac{83}{696} a^{6} + \frac{133}{696} a^{5} - \frac{3}{8} a^{4} - \frac{2}{87} a^{3} + \frac{11}{174} a^{2} - \frac{311}{696} a + \frac{8}{29}$, $\frac{1}{24876180300347050572547404221554742544} a^{17} - \frac{3434171938243875684389355072217415}{8292060100115683524182468073851580848} a^{16} + \frac{516754934312728056533123863343528653}{24876180300347050572547404221554742544} a^{15} + \frac{66366171776909065581811393306281481}{3109522537543381321568425527694342818} a^{14} + \frac{7139811656666846249107141511301761}{2073015025028920881045617018462895212} a^{13} + \frac{1925666314524564166022685556499287173}{8292060100115683524182468073851580848} a^{12} - \frac{3962889386256863785388918332272945029}{24876180300347050572547404221554742544} a^{11} + \frac{1079618857394183987213748305405466761}{24876180300347050572547404221554742544} a^{10} - \frac{3146475864778347910331447734125482575}{24876180300347050572547404221554742544} a^{9} + \frac{923744442463011297401610029625074137}{2073015025028920881045617018462895212} a^{8} + \frac{7440554462447022895266445135291181887}{24876180300347050572547404221554742544} a^{7} - \frac{1439622748003415245999714580329286819}{4146030050057841762091234036925790424} a^{6} - \frac{2822008228722096633165511172406019607}{12438090150173525286273702110777371272} a^{5} + \frac{9682946352493973252972633796749277743}{24876180300347050572547404221554742544} a^{4} + \frac{1430292509861831196367094277132925375}{8292060100115683524182468073851580848} a^{3} - \frac{9418430783254094062283018758133564887}{24876180300347050572547404221554742544} a^{2} + \frac{676302913327417128954571914340572503}{6219045075086762643136851055388685636} a + \frac{3127713707316346622314071197457938337}{8292060100115683524182468073851580848}$

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

Class group and class number

$C_{216}$, which has order $216$ (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:  \( 146431.643617 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3\times A_4$ (as 18T60):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 24 conjugacy class representatives for $C_2\times S_3\times A_4$
Character table for $C_2\times S_3\times A_4$ is not computed

Intermediate fields

3.3.148.1, \(\Q(\zeta_{7})^+\), 6.0.400967.1, 9.9.381393587008.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$37$37.6.0.1$x^{6} - x + 20$$1$$6$$0$$C_6$$[\ ]^{6}$
37.12.6.1$x^{12} + 2026120 x^{6} - 69343957 x^{2} + 1026290563600$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
167Data not computed