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Properties

Label 18.2.81919392249...1097.2
Degree $18$
Signature $[2, 8]$
Discriminant $7^{12}\cdot 41^{3}\cdot 97^{5}$
Root discriminant $24.21$
Ramified primes $7, 41, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T840

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125, -1450, -155, -1768, 619, 130, 127, -300, -286, 46, 80, -68, 27, -62, 35, -16, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 8*x^16 - 16*x^15 + 35*x^14 - 62*x^13 + 27*x^12 - 68*x^11 + 80*x^10 + 46*x^9 - 286*x^8 - 300*x^7 + 127*x^6 + 130*x^5 + 619*x^4 - 1768*x^3 - 155*x^2 - 1450*x + 125)
 
gp: K = bnfinit(x^18 - 4*x^17 + 8*x^16 - 16*x^15 + 35*x^14 - 62*x^13 + 27*x^12 - 68*x^11 + 80*x^10 + 46*x^9 - 286*x^8 - 300*x^7 + 127*x^6 + 130*x^5 + 619*x^4 - 1768*x^3 - 155*x^2 - 1450*x + 125, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 8 x^{16} - 16 x^{15} + 35 x^{14} - 62 x^{13} + 27 x^{12} - 68 x^{11} + 80 x^{10} + 46 x^{9} - 286 x^{8} - 300 x^{7} + 127 x^{6} + 130 x^{5} + 619 x^{4} - 1768 x^{3} - 155 x^{2} - 1450 x + 125 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8191939224918986549431097=7^{12}\cdot 41^{3}\cdot 97^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.21$
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:  $7, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{16} + \frac{1}{10} a^{15} - \frac{1}{5} a^{14} - \frac{1}{10} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{2} a^{8} - \frac{2}{5} a^{7} - \frac{1}{10} a^{6} - \frac{3}{10} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{633227766627584581065389650} a^{17} - \frac{7406913605816105892417439}{633227766627584581065389650} a^{16} + \frac{7388582902826553551870024}{316613883313792290532694825} a^{15} + \frac{48283077505976023346312602}{316613883313792290532694825} a^{14} + \frac{6620242358469476970935757}{63322776662758458106538965} a^{13} - \frac{47108872317843668196517937}{633227766627584581065389650} a^{12} + \frac{9436055351604586947882036}{316613883313792290532694825} a^{11} - \frac{123657317740707337043287863}{633227766627584581065389650} a^{10} - \frac{7009353829012514161289234}{63322776662758458106538965} a^{9} + \frac{57377179046630154931991821}{633227766627584581065389650} a^{8} + \frac{276311531534110162013760429}{633227766627584581065389650} a^{7} + \frac{7163491356697499312416527}{126645553325516916213077930} a^{6} + \frac{156895593011572925481458827}{633227766627584581065389650} a^{5} + \frac{6542182129945640391597116}{63322776662758458106538965} a^{4} - \frac{46218261938922342969616928}{316613883313792290532694825} a^{3} + \frac{41248696049264328721372546}{316613883313792290532694825} a^{2} + \frac{11477651248160658833286213}{25329110665103383242615586} a + \frac{990431634497319993150892}{12664555332551691621307793}$

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:  $9$
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:  \( 49001.4623877 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T840:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 165888
The 180 conjugacy class representatives for t18n840 are not computed
Character table for t18n840 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.5.467890073.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
41Data not computed
97Data not computed