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Properties

Label 18.0.53542251993...4671.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 13^{15}$
Root discriminant $30.54$
Ramified primes $3, 13$
Class number $4$
Class group $[4]$
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6241, -3555, 6084, -7431, 18516, -5124, 6151, -768, 4785, -1525, 1596, -717, 865, -495, 309, -107, 36, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 36*x^16 - 107*x^15 + 309*x^14 - 495*x^13 + 865*x^12 - 717*x^11 + 1596*x^10 - 1525*x^9 + 4785*x^8 - 768*x^7 + 6151*x^6 - 5124*x^5 + 18516*x^4 - 7431*x^3 + 6084*x^2 - 3555*x + 6241)
 
gp: K = bnfinit(x^18 - 6*x^17 + 36*x^16 - 107*x^15 + 309*x^14 - 495*x^13 + 865*x^12 - 717*x^11 + 1596*x^10 - 1525*x^9 + 4785*x^8 - 768*x^7 + 6151*x^6 - 5124*x^5 + 18516*x^4 - 7431*x^3 + 6084*x^2 - 3555*x + 6241, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 36 x^{16} - 107 x^{15} + 309 x^{14} - 495 x^{13} + 865 x^{12} - 717 x^{11} + 1596 x^{10} - 1525 x^{9} + 4785 x^{8} - 768 x^{7} + 6151 x^{6} - 5124 x^{5} + 18516 x^{4} - 7431 x^{3} + 6084 x^{2} - 3555 x + 6241 \)

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:  \(-535422519938359574117644671=-\,3^{21}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.54$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{12} + \frac{1}{16} a^{10} + \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{16} a^{5} - \frac{3}{16} a^{4} + \frac{5}{16} a^{3} + \frac{1}{8} a^{2} + \frac{5}{16} a + \frac{3}{16}$, $\frac{1}{16} a^{13} + \frac{1}{16} a^{11} + \frac{1}{16} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{16} a^{6} - \frac{3}{16} a^{5} - \frac{3}{16} a^{4} + \frac{1}{8} a^{3} - \frac{3}{16} a^{2} - \frac{5}{16} a - \frac{1}{2}$, $\frac{1}{16} a^{14} + \frac{1}{16} a^{11} - \frac{1}{16} a^{10} + \frac{1}{16} a^{9} + \frac{1}{16} a^{7} + \frac{1}{16} a^{6} + \frac{1}{8} a^{5} + \frac{1}{16} a^{4} - \frac{1}{4} a^{3} + \frac{5}{16} a^{2} + \frac{7}{16} a + \frac{1}{16}$, $\frac{1}{160} a^{15} - \frac{3}{160} a^{14} + \frac{3}{160} a^{13} - \frac{3}{160} a^{12} + \frac{19}{160} a^{11} + \frac{3}{160} a^{10} + \frac{9}{160} a^{9} - \frac{1}{32} a^{8} + \frac{7}{80} a^{7} - \frac{9}{40} a^{6} + \frac{13}{80} a^{5} - \frac{1}{40} a^{4} - \frac{73}{160} a^{3} - \frac{49}{160} a^{2} + \frac{9}{160} a + \frac{33}{160}$, $\frac{1}{12640} a^{16} - \frac{1}{1264} a^{15} + \frac{117}{6320} a^{14} - \frac{87}{6320} a^{13} - \frac{9}{632} a^{12} + \frac{141}{1264} a^{11} - \frac{17}{790} a^{10} + \frac{541}{6320} a^{9} + \frac{349}{12640} a^{8} - \frac{671}{3160} a^{7} + \frac{1559}{6320} a^{6} + \frac{83}{1580} a^{5} + \frac{383}{2528} a^{4} + \frac{1271}{6320} a^{3} + \frac{33}{3160} a^{2} - \frac{551}{1264} a + \frac{41}{160}$, $\frac{1}{313808538557592562167315680} a^{17} - \frac{2371816746330192519271}{313808538557592562167315680} a^{16} + \frac{108734358036895169617049}{78452134639398140541828920} a^{15} + \frac{308108575278443653576823}{156904269278796281083657840} a^{14} + \frac{820725684611754020181487}{31380853855759256216731568} a^{13} + \frac{196422735497774019712372}{9806516829924767567728615} a^{12} + \frac{1486762588294262042035401}{19613033659849535135457230} a^{11} + \frac{1458488338007155505546925}{31380853855759256216731568} a^{10} + \frac{12078379722434239779831}{105837618400537120461152} a^{9} - \frac{21414374679809416540123573}{313808538557592562167315680} a^{8} + \frac{353266365111712571545351}{15690426927879628108365784} a^{7} + \frac{3573377043728928701342071}{78452134639398140541828920} a^{6} + \frac{42160616292575141253568753}{313808538557592562167315680} a^{5} - \frac{13240802887097285429011821}{313808538557592562167315680} a^{4} - \frac{14495854817341416393451717}{39226067319699070270914460} a^{3} + \frac{1961477181390767820248705}{15690426927879628108365784} a^{2} + \frac{121989949987028705868035837}{313808538557592562167315680} a + \frac{1631685016543961144241783}{3972259981741678002117920}$

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

Class group and class number

$C_{4}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 327986.481487 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-39}) \), 3.1.351.1 x3, 3.3.169.1, 6.0.4804839.1, 6.0.10024911.1, 6.0.812017791.1 x2, 9.3.1235079060111.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.812017791.1
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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$