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Properties

Label 18.6.64340978779...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{8}\cdot 3^{30}\cdot 5^{13}$
Root discriminant $27.15$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_3^2:S_3$ (as 18T52)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36, 0, -108, -642, -72, 2646, -2377, 3867, -2505, 806, -948, 516, -276, 120, -33, -19, 12, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 12*x^16 - 19*x^15 - 33*x^14 + 120*x^13 - 276*x^12 + 516*x^11 - 948*x^10 + 806*x^9 - 2505*x^8 + 3867*x^7 - 2377*x^6 + 2646*x^5 - 72*x^4 - 642*x^3 - 108*x^2 + 36)
 
gp: K = bnfinit(x^18 - 3*x^17 + 12*x^16 - 19*x^15 - 33*x^14 + 120*x^13 - 276*x^12 + 516*x^11 - 948*x^10 + 806*x^9 - 2505*x^8 + 3867*x^7 - 2377*x^6 + 2646*x^5 - 72*x^4 - 642*x^3 - 108*x^2 + 36, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 12 x^{16} - 19 x^{15} - 33 x^{14} + 120 x^{13} - 276 x^{12} + 516 x^{11} - 948 x^{10} + 806 x^{9} - 2505 x^{8} + 3867 x^{7} - 2377 x^{6} + 2646 x^{5} - 72 x^{4} - 642 x^{3} - 108 x^{2} + 36 \)

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:  \(64340978779577812500000000=2^{8}\cdot 3^{30}\cdot 5^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.15$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{3}$, $\frac{1}{6} a^{16} - \frac{1}{6} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{6} a^{7} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{83745987252811506891887387655246} a^{17} + \frac{3354338197214048338787470019579}{41872993626405753445943693827623} a^{16} - \frac{869082846782361026735677755277}{83745987252811506891887387655246} a^{15} + \frac{46025728583289428714166731761}{2887792663890051961789220263974} a^{14} - \frac{3733548307602431310037421647046}{41872993626405753445943693827623} a^{13} - \frac{13772465700501625765469867483495}{83745987252811506891887387655246} a^{12} - \frac{1783169620949214197698767413260}{13957664542135251148647897942541} a^{11} + \frac{1438271869676310666012010258142}{13957664542135251148647897942541} a^{10} + \frac{4844585294458489033492760773668}{13957664542135251148647897942541} a^{9} - \frac{19128503407074090465257920707875}{41872993626405753445943693827623} a^{8} - \frac{24845771966410306626086501001169}{83745987252811506891887387655246} a^{7} + \frac{15804094173693168880904175831317}{41872993626405753445943693827623} a^{6} + \frac{17739179660871358337947171435741}{41872993626405753445943693827623} a^{5} - \frac{27665775559488423190008758490205}{83745987252811506891887387655246} a^{4} - \frac{596141780427084331309291641079}{41872993626405753445943693827623} a^{3} + \frac{6120023740883440645435780784976}{13957664542135251148647897942541} a^{2} + \frac{152457813905997950674049847981}{13957664542135251148647897942541} a + \frac{2218505545856268720815670068057}{13957664542135251148647897942541}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 1849165.3771219314 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3^2:S_3$ (as 18T52):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times C_3^2:S_3$
Character table for $C_2\times C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.243.1, 6.2.7381125.1, 9.3.143489070000.1

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.10.2$x^{6} + 9$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
3.6.10.2$x^{6} + 9$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
3.6.10.2$x^{6} + 9$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.10.2$x^{12} + 15 x^{6} + 100$$6$$2$$10$$C_6\times S_3$$[\ ]_{6}^{6}$