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Properties

Label 18.0.14851722962...8368.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{30}\cdot 7^{15}\cdot 79^{15}$
Root discriminant $612.80$
Ramified primes $2, 7, 79$
Class number $1000719360$ (GRH)
Class group $[2, 2, 2, 2, 4, 12, 1303020]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![692684296192, 0, 20090550387600, 0, 6348478632580, 0, 720817500956, 0, 37101053689, 0, 993573441, 0, 14852474, 0, 124978, 0, 553, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 553*x^16 + 124978*x^14 + 14852474*x^12 + 993573441*x^10 + 37101053689*x^8 + 720817500956*x^6 + 6348478632580*x^4 + 20090550387600*x^2 + 692684296192)
 
gp: K = bnfinit(x^18 + 553*x^16 + 124978*x^14 + 14852474*x^12 + 993573441*x^10 + 37101053689*x^8 + 720817500956*x^6 + 6348478632580*x^4 + 20090550387600*x^2 + 692684296192, 1)
 

Normalized defining polynomial

\( x^{18} + 553 x^{16} + 124978 x^{14} + 14852474 x^{12} + 993573441 x^{10} + 37101053689 x^{8} + 720817500956 x^{6} + 6348478632580 x^{4} + 20090550387600 x^{2} + 692684296192 \)

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:  \(-148517229624401636719219621434603498075107117498368=-\,2^{30}\cdot 7^{15}\cdot 79^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $612.80$
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:  $2, 7, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{1106} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2212} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{2212} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{4424} a^{9} - \frac{1}{4424} a^{8} - \frac{1}{2212} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{137144} a^{10} - \frac{1}{137144} a^{8} + \frac{41}{137144} a^{6} - \frac{55}{248} a^{4} - \frac{5}{124} a^{2} - \frac{11}{31}$, $\frac{1}{274288} a^{11} - \frac{1}{274288} a^{9} - \frac{1}{4424} a^{8} + \frac{41}{274288} a^{7} - \frac{1}{2212} a^{6} - \frac{55}{496} a^{5} + \frac{1}{8} a^{4} - \frac{5}{248} a^{3} + \frac{1}{4} a^{2} + \frac{10}{31} a$, $\frac{1}{303362528} a^{12} - \frac{1}{548576} a^{11} + \frac{1}{548576} a^{10} - \frac{61}{548576} a^{9} + \frac{53}{548576} a^{8} - \frac{41}{548576} a^{7} - \frac{233}{548576} a^{6} + \frac{117}{992} a^{5} - \frac{83}{496} a^{4} - \frac{119}{496} a^{3} - \frac{9}{31} a^{2} + \frac{11}{124} a + \frac{4}{31}$, $\frac{1}{303362528} a^{13} - \frac{1}{68572} a^{9} - \frac{13}{274288} a^{7} - \frac{49}{992} a^{5} - \frac{139}{496} a^{3} - \frac{35}{124} a$, $\frac{1}{18808476736} a^{14} - \frac{13}{9404238368} a^{12} - \frac{29}{17005856} a^{10} + \frac{27}{8502928} a^{8} - \frac{10119}{34011712} a^{6} - \frac{1041}{30752} a^{4} - \frac{1}{2} a^{3} + \frac{187}{7688} a^{2} - \frac{1}{2} a - \frac{453}{961}$, $\frac{1}{37616953472} a^{15} + \frac{9}{9404238368} a^{13} + \frac{33}{34011712} a^{11} + \frac{897}{8502928} a^{9} - \frac{1}{4424} a^{8} + \frac{1247}{9717632} a^{7} - \frac{1}{2212} a^{6} + \frac{2781}{30752} a^{5} + \frac{1}{8} a^{4} - \frac{401}{30752} a^{3} + \frac{1}{4} a^{2} - \frac{1657}{7688} a$, $\frac{1}{78807440371468428928} a^{16} - \frac{506551}{45395990997389648} a^{14} + \frac{21500189059}{39403720185734214464} a^{12} - \frac{1}{548576} a^{11} + \frac{44982768231}{35627233440989344} a^{10} - \frac{61}{548576} a^{9} - \frac{19043521715963}{142508933763957376} a^{8} - \frac{41}{548576} a^{7} - \frac{5659119497033}{35627233440989344} a^{6} + \frac{117}{992} a^{5} - \frac{9491400279561}{64425376927648} a^{4} - \frac{119}{496} a^{3} + \frac{672638349803}{16106344231912} a^{2} + \frac{11}{124} a - \frac{846360406444}{2013293028989}$, $\frac{1}{157614880742936857856} a^{17} + \frac{39213723}{5084350991707640576} a^{15} + \frac{8458593163}{11258205767352632704} a^{13} - \frac{230569149735}{142508933763957376} a^{11} - \frac{19336821428823}{285017867527914752} a^{9} - \frac{1}{4424} a^{8} + \frac{28123227318189}{285017867527914752} a^{7} - \frac{1}{2212} a^{6} - \frac{21135398880103}{128850753855296} a^{5} + \frac{1}{8} a^{4} + \frac{36011495778057}{128850753855296} a^{3} + \frac{1}{4} a^{2} - \frac{5955929049391}{32212688463824} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{12}\times C_{1303020}$, which has order $1000719360$ (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:  \( 20721910881.979282 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-553}) \), 3.3.305809.1, 3.3.49928.1, Deg 6, 6.0.3309829561471552.1, 9.9.14642685979950146048.3

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
7Data not computed
79Data not computed