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Properties

Label 18.2.11351585665...0000.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{8}\cdot 5^{15}\cdot 7^{12}$
Root discriminant $36.19$
Ramified primes $2, 3, 5, 7$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times C_3:S_3$ (as 18T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20, -480, 280, -640, 1110, -2660, 1899, 831, -2977, 2090, 175, -1479, 1134, -443, 45, 20, -7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 7*x^16 + 20*x^15 + 45*x^14 - 443*x^13 + 1134*x^12 - 1479*x^11 + 175*x^10 + 2090*x^9 - 2977*x^8 + 831*x^7 + 1899*x^6 - 2660*x^5 + 1110*x^4 - 640*x^3 + 280*x^2 - 480*x + 20)
 
gp: K = bnfinit(x^18 - 3*x^17 - 7*x^16 + 20*x^15 + 45*x^14 - 443*x^13 + 1134*x^12 - 1479*x^11 + 175*x^10 + 2090*x^9 - 2977*x^8 + 831*x^7 + 1899*x^6 - 2660*x^5 + 1110*x^4 - 640*x^3 + 280*x^2 - 480*x + 20, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 7 x^{16} + 20 x^{15} + 45 x^{14} - 443 x^{13} + 1134 x^{12} - 1479 x^{11} + 175 x^{10} + 2090 x^{9} - 2977 x^{8} + 831 x^{7} + 1899 x^{6} - 2660 x^{5} + 1110 x^{4} - 640 x^{3} + 280 x^{2} - 480 x + 20 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11351585665720125000000000000=2^{12}\cdot 3^{8}\cdot 5^{15}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.19$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} + \frac{1}{3} a^{10} - \frac{1}{2} a^{9} + \frac{1}{6} a^{6} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{13} + \frac{1}{3} a^{11} - \frac{1}{2} a^{10} + \frac{1}{6} a^{7} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{12} a^{14} + \frac{1}{4} a^{11} - \frac{1}{3} a^{10} + \frac{1}{12} a^{8} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} + \frac{1}{4} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{12} a^{9} - \frac{1}{2} a^{8} - \frac{1}{6} a^{7} - \frac{1}{12} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{4399956} a^{16} - \frac{25661}{4399956} a^{15} + \frac{25}{199998} a^{14} - \frac{163391}{4399956} a^{13} - \frac{119257}{4399956} a^{12} - \frac{621829}{2199978} a^{11} + \frac{5861}{35772} a^{10} + \frac{949555}{4399956} a^{9} - \frac{1028561}{2199978} a^{8} - \frac{160391}{399996} a^{7} + \frac{499747}{1466652} a^{6} - \frac{814469}{2199978} a^{5} - \frac{755423}{2199978} a^{4} - \frac{34942}{1099989} a^{3} - \frac{157853}{1099989} a^{2} + \frac{120557}{1099989} a + \frac{299308}{1099989}$, $\frac{1}{14893532123001712676892} a^{17} + \frac{73441387685535}{1654836902555745852988} a^{16} - \frac{84228955527149209837}{2482255353833618779482} a^{15} + \frac{11101300018077798091}{827418451277872926494} a^{14} - \frac{514558475573937528521}{14893532123001712676892} a^{13} + \frac{478541608267056655081}{7446766061500856338446} a^{12} - \frac{2953268574596053633739}{7446766061500856338446} a^{11} + \frac{2533677943887537547327}{14893532123001712676892} a^{10} + \frac{3703106631942880240013}{7446766061500856338446} a^{9} - \frac{109608772747220926747}{1241127676916809389741} a^{8} - \frac{4467936919404115744481}{14893532123001712676892} a^{7} - \frac{647111711077479946864}{3723383030750428169223} a^{6} - \frac{920167461189600596645}{4964510707667237558964} a^{5} - \frac{900883223974351668559}{2482255353833618779482} a^{4} + \frac{2727286255051732066999}{7446766061500856338446} a^{3} + \frac{675528246489319088807}{7446766061500856338446} a^{2} + \frac{1663343208270350643530}{3723383030750428169223} a - \frac{49109712408675667873}{3723383030750428169223}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $9$
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:  \( 7580353.718580581 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_3$ (as 18T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 12 conjugacy class representatives for $C_2\times C_3:S_3$
Character table for $C_2\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.300.1, 3.1.3675.1, 3.1.14700.1, 3.1.588.1, 6.2.450000.1, 6.2.1080450000.1, 6.2.67528125.1, 6.2.43218000.1, 9.1.9529569000000.1

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 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 R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
7Data not computed