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Properties

Label 18.0.47750353976...3399.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{37}\cdot 13^{9}$
Root discriminant $34.49$
Ramified primes $3, 13$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3136, 6048, 1296, -6792, 13860, -5994, 11457, -9792, 7209, -2056, 2241, -972, 399, -72, 81, 0, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 + 81*x^14 - 72*x^13 + 399*x^12 - 972*x^11 + 2241*x^10 - 2056*x^9 + 7209*x^8 - 9792*x^7 + 11457*x^6 - 5994*x^5 + 13860*x^4 - 6792*x^3 + 1296*x^2 + 6048*x + 3136)
 
gp: K = bnfinit(x^18 + 9*x^16 + 81*x^14 - 72*x^13 + 399*x^12 - 972*x^11 + 2241*x^10 - 2056*x^9 + 7209*x^8 - 9792*x^7 + 11457*x^6 - 5994*x^5 + 13860*x^4 - 6792*x^3 + 1296*x^2 + 6048*x + 3136, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} + 81 x^{14} - 72 x^{13} + 399 x^{12} - 972 x^{11} + 2241 x^{10} - 2056 x^{9} + 7209 x^{8} - 9792 x^{7} + 11457 x^{6} - 5994 x^{5} + 13860 x^{4} - 6792 x^{3} + 1296 x^{2} + 6048 x + 3136 \)

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:  \(-4775035397693072542278153399=-\,3^{37}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.49$
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:  $3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{3} - \frac{1}{3} a$, $\frac{1}{18} a^{11} - \frac{1}{18} a^{10} + \frac{1}{18} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{18} a^{2} + \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{234} a^{12} + \frac{5}{78} a^{10} + \frac{2}{117} a^{9} + \frac{1}{39} a^{8} - \frac{11}{78} a^{5} + \frac{5}{13} a^{4} + \frac{37}{234} a^{3} - \frac{2}{39} a^{2} + \frac{16}{39} a + \frac{2}{117}$, $\frac{1}{234} a^{13} + \frac{1}{117} a^{11} + \frac{17}{234} a^{10} - \frac{7}{234} a^{9} - \frac{11}{78} a^{6} + \frac{5}{13} a^{5} - \frac{40}{117} a^{4} + \frac{35}{78} a^{3} + \frac{83}{234} a^{2} - \frac{50}{117} a + \frac{4}{9}$, $\frac{1}{468} a^{14} - \frac{1}{468} a^{12} + \frac{1}{117} a^{11} - \frac{1}{12} a^{10} + \frac{7}{234} a^{9} + \frac{7}{156} a^{8} + \frac{1}{78} a^{7} + \frac{17}{156} a^{6} - \frac{5}{117} a^{5} + \frac{49}{156} a^{4} - \frac{53}{234} a^{3} - \frac{155}{468} a^{2} + \frac{17}{78} a + \frac{23}{117}$, $\frac{1}{85176} a^{15} - \frac{1}{1092} a^{14} + \frac{181}{85176} a^{13} + \frac{3}{4732} a^{12} - \frac{19}{728} a^{11} + \frac{1643}{42588} a^{10} + \frac{239}{6552} a^{9} - \frac{781}{14196} a^{8} - \frac{373}{28392} a^{7} - \frac{2545}{42588} a^{6} + \frac{3421}{9464} a^{5} + \frac{1553}{3276} a^{4} + \frac{1943}{28392} a^{3} + \frac{3007}{7098} a^{2} - \frac{3841}{10647} a + \frac{349}{1521}$, $\frac{1}{511056} a^{16} - \frac{1}{255528} a^{15} + \frac{7}{8112} a^{14} + \frac{535}{255528} a^{13} + \frac{425}{511056} a^{12} + \frac{23}{1352} a^{11} + \frac{17335}{511056} a^{10} + \frac{2519}{255528} a^{9} + \frac{1571}{18928} a^{8} + \frac{8987}{255528} a^{7} + \frac{53449}{511056} a^{6} - \frac{12469}{28392} a^{5} - \frac{50383}{511056} a^{4} - \frac{40297}{127764} a^{3} - \frac{516}{1183} a^{2} + \frac{6641}{63882} a - \frac{2273}{4563}$, $\frac{1}{229155552993171168} a^{17} - \frac{2895173709}{4243621351725392} a^{16} - \frac{147753606667}{229155552993171168} a^{15} + \frac{76956455119}{1259096445017424} a^{14} + \frac{11114501063335}{8487242703450784} a^{13} + \frac{37240543445081}{114577776496585584} a^{12} + \frac{369419096187805}{32736507570453024} a^{11} + \frac{23185505684155}{1818694865025168} a^{10} + \frac{1429466438447393}{17627350230243936} a^{9} + \frac{494362348142297}{114577776496585584} a^{8} + \frac{529609410492989}{25461728110352352} a^{7} + \frac{11824785371227669}{114577776496585584} a^{6} - \frac{75164446669711123}{229155552993171168} a^{5} - \frac{196261899537815}{1591358006897022} a^{4} - \frac{291883505245697}{784779291072504} a^{3} - \frac{6934281183290795}{14322222062073198} a^{2} - \frac{274571143913314}{795679003448511} a + \frac{228174108655813}{1023015861576657}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 2302656.90616 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-39}) \), 3.1.3159.2 x3, 3.1.3159.1 x3, 3.1.351.1 x3, 3.1.3159.3 x3, 6.0.389191959.4, 6.0.389191959.6, 6.0.4804839.1, 6.0.389191959.1, 9.1.11065116586329.3 x9

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

Sibling fields

Degree 9 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$