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Properties

Label 18.0.54091770075...1408.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{18}\cdot 7^{14}\cdot 43^{9}$
Root discriminant $141.85$
Ramified primes $2, 3, 7, 43$
Class number $2108592$ (GRH)
Class group $[4, 527148]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![33569437537, -18328267323, 15888370065, -6011436698, 3196420548, -998800626, 414527356, -111524946, 37842855, -8744095, 2517105, -504282, 124098, -20712, 4272, -572, 99, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 99*x^16 - 572*x^15 + 4272*x^14 - 20712*x^13 + 124098*x^12 - 504282*x^11 + 2517105*x^10 - 8744095*x^9 + 37842855*x^8 - 111524946*x^7 + 414527356*x^6 - 998800626*x^5 + 3196420548*x^4 - 6011436698*x^3 + 15888370065*x^2 - 18328267323*x + 33569437537)
 
gp: K = bnfinit(x^18 - 9*x^17 + 99*x^16 - 572*x^15 + 4272*x^14 - 20712*x^13 + 124098*x^12 - 504282*x^11 + 2517105*x^10 - 8744095*x^9 + 37842855*x^8 - 111524946*x^7 + 414527356*x^6 - 998800626*x^5 + 3196420548*x^4 - 6011436698*x^3 + 15888370065*x^2 - 18328267323*x + 33569437537, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 99 x^{16} - 572 x^{15} + 4272 x^{14} - 20712 x^{13} + 124098 x^{12} - 504282 x^{11} + 2517105 x^{10} - 8744095 x^{9} + 37842855 x^{8} - 111524946 x^{7} + 414527356 x^{6} - 998800626 x^{5} + 3196420548 x^{4} - 6011436698 x^{3} + 15888370065 x^{2} - 18328267323 x + 33569437537 \)

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:  \(-540917700754627690421790767951711121408=-\,2^{12}\cdot 3^{18}\cdot 7^{14}\cdot 43^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $141.85$
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:  $2, 3, 7, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4}$, $\frac{1}{143514756374606089349970882012300586220330738964413265668554532} a^{17} - \frac{463262827050671169821698568880433175648280645723140276841470}{35878689093651522337492720503075146555082684741103316417138633} a^{16} + \frac{6044672267680269600947185434735366308173955379763330721116653}{71757378187303044674985441006150293110165369482206632834277266} a^{15} - \frac{1952253066731839768354351195902782658233515482278096764698845}{71757378187303044674985441006150293110165369482206632834277266} a^{14} + \frac{4226516061112013713396511525095879423874394508747423143863583}{35878689093651522337492720503075146555082684741103316417138633} a^{13} - \frac{1735637188916809412595666974322335939538181176093241201052541}{71757378187303044674985441006150293110165369482206632834277266} a^{12} + \frac{7850651206403645148290473584952902213707145046307399194855060}{35878689093651522337492720503075146555082684741103316417138633} a^{11} - \frac{12273048839629215212244576128848365890297933452043759499730669}{71757378187303044674985441006150293110165369482206632834277266} a^{10} - \frac{16573002416726523328754227140108525489012531631297072058114449}{143514756374606089349970882012300586220330738964413265668554532} a^{9} + \frac{2975631263108586233949421920994871783371267143429122659221601}{71757378187303044674985441006150293110165369482206632834277266} a^{8} - \frac{1281395991697884485480254779262326181795532861400298095815031}{5519798322100234205768110846626945623858874575554356371867482} a^{7} + \frac{79437979906168051261398467884439213180691574592472852230087}{5519798322100234205768110846626945623858874575554356371867482} a^{6} + \frac{6383158855393326582829134068603348684089326130055785675067497}{35878689093651522337492720503075146555082684741103316417138633} a^{5} - \frac{10925115756080383532351405269817053567026639280085450683238301}{35878689093651522337492720503075146555082684741103316417138633} a^{4} + \frac{11873113981562145047301238359359032814990763744347461908727701}{35878689093651522337492720503075146555082684741103316417138633} a^{3} - \frac{1480750831656687123104830965317950269106930898383604056152633}{5519798322100234205768110846626945623858874575554356371867482} a^{2} + \frac{47099790740752206843428217667115511124932211722263956431565235}{143514756374606089349970882012300586220330738964413265668554532} a - \frac{30914842177817440983136662420464651284811030773141835374248073}{71757378187303044674985441006150293110165369482206632834277266}$

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

Class group and class number

$C_{4}\times C_{527148}$, which has order $2108592$ (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

\(\Q(\sqrt{-43}) \), \(\Q(\zeta_{7})^+\), 3.3.756.1, 6.0.190896307.1, 6.0.45441112752.2, 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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
2Data not computed
3Data not computed
$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}$
$43$43.6.3.2$x^{6} - 1849 x^{2} + 795070$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
43.6.3.2$x^{6} - 1849 x^{2} + 795070$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
43.6.3.2$x^{6} - 1849 x^{2} + 795070$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$