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Properties

Label 18.18.6764488804...7389.1
Degree $18$
Signature $[18, 0]$
Discriminant $61^{3}\cdot 1129^{9}$
Root discriminant $66.67$
Ramified primes $61, 1129$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times C_2^2:D_9$ (as 18T67)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -155, 2705, -302, -25638, 8710, 76097, -25896, -87762, 21928, 47140, -6240, -12078, 413, 1313, -7, -61, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 61*x^16 - 7*x^15 + 1313*x^14 + 413*x^13 - 12078*x^12 - 6240*x^11 + 47140*x^10 + 21928*x^9 - 87762*x^8 - 25896*x^7 + 76097*x^6 + 8710*x^5 - 25638*x^4 - 302*x^3 + 2705*x^2 - 155*x + 1)
 
gp: K = bnfinit(x^18 - 61*x^16 - 7*x^15 + 1313*x^14 + 413*x^13 - 12078*x^12 - 6240*x^11 + 47140*x^10 + 21928*x^9 - 87762*x^8 - 25896*x^7 + 76097*x^6 + 8710*x^5 - 25638*x^4 - 302*x^3 + 2705*x^2 - 155*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 61 x^{16} - 7 x^{15} + 1313 x^{14} + 413 x^{13} - 12078 x^{12} - 6240 x^{11} + 47140 x^{10} + 21928 x^{9} - 87762 x^{8} - 25896 x^{7} + 76097 x^{6} + 8710 x^{5} - 25638 x^{4} - 302 x^{3} + 2705 x^{2} - 155 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(676448880473573509966951229297389=61^{3}\cdot 1129^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.67$
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:  $61, 1129$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{33} a^{16} - \frac{5}{33} a^{15} - \frac{7}{33} a^{14} + \frac{5}{11} a^{13} + \frac{4}{11} a^{12} - \frac{4}{33} a^{11} + \frac{5}{33} a^{10} - \frac{4}{11} a^{9} - \frac{10}{33} a^{8} + \frac{5}{11} a^{7} + \frac{16}{33} a^{6} + \frac{1}{33} a^{5} - \frac{4}{33} a^{4} + \frac{14}{33} a^{3} + \frac{5}{11} a^{2} - \frac{4}{33} a - \frac{8}{33}$, $\frac{1}{366211228480297973472869537336397} a^{17} + \frac{5349806615973234827954524460024}{366211228480297973472869537336397} a^{16} + \frac{42107449838530861576700677841312}{122070409493432657824289845778799} a^{15} - \frac{179700856987429546208507773492555}{366211228480297973472869537336397} a^{14} - \frac{16628978918760725587689784397693}{40690136497810885941429948592933} a^{13} + \frac{11720648276359370646246966125051}{366211228480297973472869537336397} a^{12} + \frac{151357229523141129659701533371596}{366211228480297973472869537336397} a^{11} + \frac{170232154293449592424712153980478}{366211228480297973472869537336397} a^{10} + \frac{25866190637878776941313768633653}{366211228480297973472869537336397} a^{9} - \frac{148368081874711734334305929126341}{366211228480297973472869537336397} a^{8} + \frac{75481186440363901124235960333442}{366211228480297973472869537336397} a^{7} + \frac{11908837420017467606314288189945}{33291929861845270315715412485127} a^{6} - \frac{27148354335116520525141597962924}{122070409493432657824289845778799} a^{5} - \frac{52040000034655265751936983057579}{366211228480297973472869537336397} a^{4} + \frac{13594001024184534185532755583749}{28170094498484459497913041333569} a^{3} + \frac{84791810446454188494388648009304}{366211228480297973472869537336397} a^{2} - \frac{42782714349485415120757202483825}{122070409493432657824289845778799} a + \frac{69763554723666012755337937053310}{366211228480297973472869537336397}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $17$
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:  \( 24555150983.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:D_9$ (as 18T67):

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 $C_2\times C_2^2:D_9$
Character table for $C_2\times C_2^2:D_9$

Intermediate fields

3.3.1129.1, 6.6.87783251029.1, 9.9.1624709678881.1

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 $18$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
61Data not computed
1129Data not computed