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Properties

Label 18.0.34211896605...0048.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 41^{8}$
Root discriminant $29.79$
Ramified primes $2, 3, 41$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![841, -2958, 8751, -12948, 18666, -14751, 17236, -9576, 15204, -2941, 6411, -2742, 943, -525, 165, -59, 12, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 12*x^16 - 59*x^15 + 165*x^14 - 525*x^13 + 943*x^12 - 2742*x^11 + 6411*x^10 - 2941*x^9 + 15204*x^8 - 9576*x^7 + 17236*x^6 - 14751*x^5 + 18666*x^4 - 12948*x^3 + 8751*x^2 - 2958*x + 841)
 
gp: K = bnfinit(x^18 - 3*x^17 + 12*x^16 - 59*x^15 + 165*x^14 - 525*x^13 + 943*x^12 - 2742*x^11 + 6411*x^10 - 2941*x^9 + 15204*x^8 - 9576*x^7 + 17236*x^6 - 14751*x^5 + 18666*x^4 - 12948*x^3 + 8751*x^2 - 2958*x + 841, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 12 x^{16} - 59 x^{15} + 165 x^{14} - 525 x^{13} + 943 x^{12} - 2742 x^{11} + 6411 x^{10} - 2941 x^{9} + 15204 x^{8} - 9576 x^{7} + 17236 x^{6} - 14751 x^{5} + 18666 x^{4} - 12948 x^{3} + 8751 x^{2} - 2958 x + 841 \)

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:  \(-342118966051435975575810048=-\,2^{12}\cdot 3^{21}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.79$
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:  $2, 3, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{153} a^{15} - \frac{20}{153} a^{14} - \frac{5}{153} a^{13} + \frac{25}{153} a^{12} + \frac{5}{51} a^{11} - \frac{10}{153} a^{10} + \frac{1}{9} a^{9} + \frac{14}{153} a^{8} + \frac{4}{51} a^{7} - \frac{1}{3} a^{6} + \frac{43}{153} a^{5} - \frac{1}{9} a^{4} - \frac{70}{153} a^{3} - \frac{38}{153} a^{2} - \frac{26}{153}$, $\frac{1}{11180658753} a^{16} - \frac{331150}{219228603} a^{15} + \frac{273915289}{3726886251} a^{14} + \frac{185435884}{1242295417} a^{13} - \frac{1762173925}{11180658753} a^{12} - \frac{1833061600}{11180658753} a^{11} + \frac{77541870}{1242295417} a^{10} - \frac{94280437}{3726886251} a^{9} + \frac{446087063}{1016423523} a^{8} + \frac{116390549}{338807841} a^{7} + \frac{2331508852}{11180658753} a^{6} - \frac{268456245}{1242295417} a^{5} + \frac{1767534793}{11180658753} a^{4} - \frac{1806799909}{11180658753} a^{3} - \frac{371907652}{11180658753} a^{2} - \frac{2272054106}{11180658753} a + \frac{105493009}{385539957}$, $\frac{1}{6458363231915605929300657} a^{17} - \frac{83558462500672}{6458363231915605929300657} a^{16} - \frac{11381840411499424040587}{6458363231915605929300657} a^{15} - \frac{14229403671007172651161}{6458363231915605929300657} a^{14} - \frac{200085423191136531311765}{6458363231915605929300657} a^{13} - \frac{727835857021232862329755}{6458363231915605929300657} a^{12} - \frac{334699055794728388645325}{6458363231915605929300657} a^{11} + \frac{345155064364914747445813}{6458363231915605929300657} a^{10} - \frac{424271221817710123480666}{6458363231915605929300657} a^{9} + \frac{11176217522286246765017}{65235992241571777063643} a^{8} - \frac{1636122685615172752726826}{6458363231915605929300657} a^{7} - \frac{1461475335556125577684864}{6458363231915605929300657} a^{6} - \frac{730451007236127219810784}{2152787743971868643100219} a^{5} - \frac{681535596565758066515183}{2152787743971868643100219} a^{4} + \frac{2739394744163293016077417}{6458363231915605929300657} a^{3} + \frac{1371304526971916339375425}{6458363231915605929300657} a^{2} + \frac{377207747493085874474194}{6458363231915605929300657} a - \frac{627631974832699821755}{24744686712320329231037}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( -\frac{306210358999066}{854325262135400151} a^{17} + \frac{908113657487209}{854325262135400151} a^{16} - \frac{3733314262640398}{854325262135400151} a^{15} + \frac{18166872036769294}{854325262135400151} a^{14} - \frac{16934685436550700}{284775087378466717} a^{13} + \frac{5643259710259999}{29459491797772419} a^{12} - \frac{98315842590947876}{284775087378466717} a^{11} + \frac{866801074421093302}{854325262135400151} a^{10} - \frac{1987506605054504878}{854325262135400151} a^{9} + \frac{1013457462654751877}{854325262135400151} a^{8} - \frac{5036062073861633680}{854325262135400151} a^{7} + \frac{2629360200861283346}{854325262135400151} a^{6} - \frac{2030263631941873688}{284775087378466717} a^{5} + \frac{4436535288094819417}{854325262135400151} a^{4} - \frac{6263447082297500600}{854325262135400151} a^{3} + \frac{67873116873664220}{16751475728145101} a^{2} - \frac{3172972347415155970}{854325262135400151} a + \frac{12520968744741790}{9819830599257473} \) (order $6$)
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:  \( 188948.15601729616 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 6.0.34992.1, 9.3.3559645040832.1

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

Sibling fields

Degree 9 siblings: data not computed
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$41$41.6.4.2$x^{6} - 41 x^{3} + 20172$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
41.6.4.2$x^{6} - 41 x^{3} + 20172$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
41.6.0.1$x^{6} - x + 7$$1$$6$$0$$C_6$$[\ ]^{6}$