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Properties

Label 18.6.81919392249...1097.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 41^{3}\cdot 97^{5}$
Root discriminant $24.21$
Ramified primes $7, 41, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T840

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1709, -1600, -3901, 4565, -2241, 716, 3710, -3244, 2164, -256, -772, 684, -382, 165, -15, -18, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 8*x^16 - 18*x^15 - 15*x^14 + 165*x^13 - 382*x^12 + 684*x^11 - 772*x^10 - 256*x^9 + 2164*x^8 - 3244*x^7 + 3710*x^6 + 716*x^5 - 2241*x^4 + 4565*x^3 - 3901*x^2 - 1600*x + 1709)
 
gp: K = bnfinit(x^18 - 4*x^17 + 8*x^16 - 18*x^15 - 15*x^14 + 165*x^13 - 382*x^12 + 684*x^11 - 772*x^10 - 256*x^9 + 2164*x^8 - 3244*x^7 + 3710*x^6 + 716*x^5 - 2241*x^4 + 4565*x^3 - 3901*x^2 - 1600*x + 1709, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 8 x^{16} - 18 x^{15} - 15 x^{14} + 165 x^{13} - 382 x^{12} + 684 x^{11} - 772 x^{10} - 256 x^{9} + 2164 x^{8} - 3244 x^{7} + 3710 x^{6} + 716 x^{5} - 2241 x^{4} + 4565 x^{3} - 3901 x^{2} - 1600 x + 1709 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1195629632966252023278852535989731081} a^{17} + \frac{111723988182755543844088341612167010}{1195629632966252023278852535989731081} a^{16} - \frac{342058149870565406186091948269607313}{1195629632966252023278852535989731081} a^{15} + \frac{117607997708513029069438390725851934}{1195629632966252023278852535989731081} a^{14} + \frac{139549102453847457997834882210432701}{1195629632966252023278852535989731081} a^{13} - \frac{366630742887414795433851905436344055}{1195629632966252023278852535989731081} a^{12} - \frac{532942051190017613796128975227469758}{1195629632966252023278852535989731081} a^{11} - \frac{2557622917885220975597539548412108}{6605688579924044327507472574528901} a^{10} + \frac{126242086545979940145759760243857601}{1195629632966252023278852535989731081} a^{9} - \frac{127103205925166404622955133833965809}{1195629632966252023278852535989731081} a^{8} - \frac{174532270786543430663965837644659418}{1195629632966252023278852535989731081} a^{7} - \frac{230340877934148068895977718948139149}{1195629632966252023278852535989731081} a^{6} + \frac{247767596939009004407345030577899522}{1195629632966252023278852535989731081} a^{5} + \frac{566161176470720930996790044007312683}{1195629632966252023278852535989731081} a^{4} + \frac{557438132269948467882477489469302922}{1195629632966252023278852535989731081} a^{3} + \frac{124754189855672045392330214901352110}{1195629632966252023278852535989731081} a^{2} + \frac{396326870935798194079213332701318694}{1195629632966252023278852535989731081} a + \frac{110918646874958984924530702052910704}{1195629632966252023278852535989731081}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 268458.813591 \) (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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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