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Properties

Label 18.12.1444338742...6288.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{12}\cdot 3^{37}\cdot 23^{8}$
Root discriminant $61.19$
Ramified primes $2, 3, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_3:S_4$ (as 18T66)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![76, -468, -576, 8988, -21960, 22230, -9777, -1287, 11511, -17092, 10953, -2313, -612, 711, -441, 120, 9, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 9*x^16 + 120*x^15 - 441*x^14 + 711*x^13 - 612*x^12 - 2313*x^11 + 10953*x^10 - 17092*x^9 + 11511*x^8 - 1287*x^7 - 9777*x^6 + 22230*x^5 - 21960*x^4 + 8988*x^3 - 576*x^2 - 468*x + 76)
 
gp: K = bnfinit(x^18 - 9*x^17 + 9*x^16 + 120*x^15 - 441*x^14 + 711*x^13 - 612*x^12 - 2313*x^11 + 10953*x^10 - 17092*x^9 + 11511*x^8 - 1287*x^7 - 9777*x^6 + 22230*x^5 - 21960*x^4 + 8988*x^3 - 576*x^2 - 468*x + 76, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 9 x^{16} + 120 x^{15} - 441 x^{14} + 711 x^{13} - 612 x^{12} - 2313 x^{11} + 10953 x^{10} - 17092 x^{9} + 11511 x^{8} - 1287 x^{7} - 9777 x^{6} + 22230 x^{5} - 21960 x^{4} + 8988 x^{3} - 576 x^{2} - 468 x + 76 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-144433874233348438770904326156288=-\,2^{12}\cdot 3^{37}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.19$
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:  $2, 3, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{56} a^{15} - \frac{1}{28} a^{14} - \frac{1}{14} a^{13} - \frac{1}{8} a^{12} + \frac{1}{7} a^{11} - \frac{1}{4} a^{10} + \frac{5}{56} a^{9} + \frac{3}{14} a^{8} + \frac{5}{28} a^{7} - \frac{23}{56} a^{6} - \frac{3}{7} a^{5} + \frac{1}{28} a^{4} + \frac{1}{14} a^{3} - \frac{3}{7} a^{2} + \frac{3}{14} a - \frac{5}{14}$, $\frac{1}{2128} a^{16} + \frac{15}{2128} a^{15} + \frac{93}{1064} a^{14} - \frac{61}{2128} a^{13} - \frac{223}{2128} a^{12} - \frac{121}{1064} a^{11} - \frac{191}{2128} a^{10} - \frac{155}{2128} a^{9} + \frac{79}{1064} a^{8} - \frac{117}{304} a^{7} - \frac{275}{2128} a^{6} - \frac{69}{152} a^{5} - \frac{81}{266} a^{4} - \frac{48}{133} a^{3} - \frac{155}{532} a^{2} - \frac{157}{532} a + \frac{11}{28}$, $\frac{1}{219724315823111905467632096} a^{17} + \frac{10468081712210481886147}{109862157911555952733816048} a^{16} + \frac{32846525049020397553917}{31389187974730272209661728} a^{15} - \frac{19885170776302669544004451}{219724315823111905467632096} a^{14} - \frac{7578311434812416154158851}{109862157911555952733816048} a^{13} - \frac{15706046467645840988957459}{219724315823111905467632096} a^{12} - \frac{15621772129003159990246833}{219724315823111905467632096} a^{11} - \frac{2260429127951775078187811}{27465539477888988183454012} a^{10} - \frac{9884571749191957570046825}{31389187974730272209661728} a^{9} + \frac{26748829632165100139171099}{219724315823111905467632096} a^{8} - \frac{2032609811416117093199457}{54931078955777976366908024} a^{7} - \frac{16666466732332757070031027}{219724315823111905467632096} a^{6} - \frac{1933489727308138295856177}{15694593987365136104830864} a^{5} - \frac{8785977142118703829237391}{54931078955777976366908024} a^{4} + \frac{404044046061712637664681}{2891109418725156650889896} a^{3} + \frac{3189478731836683926055065}{13732769738944494091727006} a^{2} + \frac{2202231645081038702500151}{13732769738944494091727006} a - \frac{155266014737369883699521}{413015631246450950127128}$

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:  $14$
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:  \( 21239033018.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_4$ (as 18T66):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 18 conjugacy class representatives for $C_2\times C_3:S_4$
Character table for $C_2\times C_3:S_4$

Intermediate fields

3.3.22356.2, 3.3.22356.1, 3.3.22356.3, 3.3.621.1, 6.4.1499372208.1, 9.9.6938632771983936.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$[\ ]_{3}^{2}$
3Data not computed
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$