↑

Properties

Label 18.0.19384991605...8128.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{18}\cdot 7^{14}\cdot 23^{9}$
Root discriminant $103.75$
Ramified primes $2, 3, 7, 23$
Class number $249318$ (GRH)
Class group $[3, 9, 9, 1026]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![728867672, -607457328, 489750240, -221288668, 109772868, -41853396, 19295616, -6888426, 2543820, -726875, 236595, -67602, 20478, -4572, 1032, -212, 54, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 54*x^16 - 212*x^15 + 1032*x^14 - 4572*x^13 + 20478*x^12 - 67602*x^11 + 236595*x^10 - 726875*x^9 + 2543820*x^8 - 6888426*x^7 + 19295616*x^6 - 41853396*x^5 + 109772868*x^4 - 221288668*x^3 + 489750240*x^2 - 607457328*x + 728867672)
 
gp: K = bnfinit(x^18 - 9*x^17 + 54*x^16 - 212*x^15 + 1032*x^14 - 4572*x^13 + 20478*x^12 - 67602*x^11 + 236595*x^10 - 726875*x^9 + 2543820*x^8 - 6888426*x^7 + 19295616*x^6 - 41853396*x^5 + 109772868*x^4 - 221288668*x^3 + 489750240*x^2 - 607457328*x + 728867672, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 54 x^{16} - 212 x^{15} + 1032 x^{14} - 4572 x^{13} + 20478 x^{12} - 67602 x^{11} + 236595 x^{10} - 726875 x^{9} + 2543820 x^{8} - 6888426 x^{7} + 19295616 x^{6} - 41853396 x^{5} + 109772868 x^{4} - 221288668 x^{3} + 489750240 x^{2} - 607457328 x + 728867672 \)

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:  \(-1938499160566797308196011628480688128=-\,2^{12}\cdot 3^{18}\cdot 7^{14}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $103.75$
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:  $2, 3, 7, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{6}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{7}$, $\frac{1}{52} a^{16} + \frac{5}{52} a^{15} + \frac{1}{13} a^{14} - \frac{1}{26} a^{13} + \frac{1}{13} a^{12} - \frac{3}{13} a^{11} - \frac{2}{13} a^{10} - \frac{7}{52} a^{8} - \frac{1}{4} a^{7} - \frac{3}{13} a^{6} + \frac{1}{26} a^{5} + \frac{7}{26} a^{4} + \frac{2}{13} a^{3} - \frac{6}{13} a^{2} - \frac{3}{13} a$, $\frac{1}{124894122261618758956204539903724263437887598000520884} a^{17} - \frac{63320090755100353869175304515318407403588839075513}{62447061130809379478102269951862131718943799000260442} a^{16} + \frac{7685500602554512148376343464290715789176854736859769}{124894122261618758956204539903724263437887598000520884} a^{15} + \frac{2329570898304397181613982783029568265575025658011164}{31223530565404689739051134975931065859471899500130221} a^{14} - \frac{6724894747572518239247348557034274987170813480007477}{62447061130809379478102269951862131718943799000260442} a^{13} + \frac{5186762327010323766898177691251797710715679807436471}{124894122261618758956204539903724263437887598000520884} a^{12} - \frac{5532776833692558359617604734736006162462545361166169}{31223530565404689739051134975931065859471899500130221} a^{11} - \frac{13558826472518607252733991590285749383221676430883469}{124894122261618758956204539903724263437887598000520884} a^{10} + \frac{11658932142600614399087512824969925818687907731755799}{124894122261618758956204539903724263437887598000520884} a^{9} - \frac{3376377166982835137011131525781967345462902257134679}{124894122261618758956204539903724263437887598000520884} a^{8} - \frac{12013777997735982980492715257942241218530620295783245}{124894122261618758956204539903724263437887598000520884} a^{7} - \frac{28730320870020503795423758435423423588636050244373275}{124894122261618758956204539903724263437887598000520884} a^{6} + \frac{4066071454955591483307597956132959583317149948692541}{62447061130809379478102269951862131718943799000260442} a^{5} - \frac{12709421413931887201106597531663523710796549571255613}{62447061130809379478102269951862131718943799000260442} a^{4} - \frac{9967998145057095246059080915234579977888356770578247}{31223530565404689739051134975931065859471899500130221} a^{3} + \frac{7863513301389182182091249726989098443805027611290987}{31223530565404689739051134975931065859471899500130221} a^{2} + \frac{9293381492368654723679656372035887473559130883644520}{31223530565404689739051134975931065859471899500130221} a + \frac{390707692680820586272245338164374539201105039580741}{2401810043492668441465471921225466604574761500010017}$

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

Class group and class number

$C_{3}\times C_{9}\times C_{9}\times C_{1026}$, which has order $249318$ (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:  \( 296124.35954857944 \) (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

\(\Q(\sqrt{-23}) \), 3.3.756.1, \(\Q(\zeta_{7})^+\), 6.0.6953878512.6, 6.0.29212967.1, 9.9.1037426999616.1

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 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$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}$
$3$3.9.9.1$x^{9} + 54 x^{5} + 27 x^{3} + 189$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
3.9.9.1$x^{9} + 54 x^{5} + 27 x^{3} + 189$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$23$23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.12.6.1$x^{12} + 365010 x^{6} - 6436343 x^{2} + 33308075025$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$