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Properties

Label 18.18.3637953142...4421.1
Degree $18$
Signature $[18, 0]$
Discriminant $7^{14}\cdot 67^{6}\cdot 181^{3}$
Root discriminant $43.88$
Ramified primes $7, 67, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times S_3\times A_4$ (as 18T60)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, 432, -2460, -1586, 15769, 2017, -35376, 3181, 36868, -9525, -18098, 7244, 3519, -1907, -213, 207, -9, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 - 9*x^16 + 207*x^15 - 213*x^14 - 1907*x^13 + 3519*x^12 + 7244*x^11 - 18098*x^10 - 9525*x^9 + 36868*x^8 + 3181*x^7 - 35376*x^6 + 2017*x^5 + 15769*x^4 - 1586*x^3 - 2460*x^2 + 432*x + 27)
 
gp: K = bnfinit(x^18 - 8*x^17 - 9*x^16 + 207*x^15 - 213*x^14 - 1907*x^13 + 3519*x^12 + 7244*x^11 - 18098*x^10 - 9525*x^9 + 36868*x^8 + 3181*x^7 - 35376*x^6 + 2017*x^5 + 15769*x^4 - 1586*x^3 - 2460*x^2 + 432*x + 27, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} - 9 x^{16} + 207 x^{15} - 213 x^{14} - 1907 x^{13} + 3519 x^{12} + 7244 x^{11} - 18098 x^{10} - 9525 x^{9} + 36868 x^{8} + 3181 x^{7} - 35376 x^{6} + 2017 x^{5} + 15769 x^{4} - 1586 x^{3} - 2460 x^{2} + 432 x + 27 \)

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:  \(363795314284140453335088294421=7^{14}\cdot 67^{6}\cdot 181^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.88$
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:  $7, 67, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{15} a^{15} + \frac{2}{5} a^{13} - \frac{2}{5} a^{12} - \frac{2}{5} a^{11} + \frac{4}{15} a^{10} + \frac{2}{15} a^{9} + \frac{2}{5} a^{8} + \frac{1}{15} a^{7} - \frac{4}{15} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{2}{15} a^{3} + \frac{2}{5} a^{2} + \frac{7}{15} a - \frac{1}{5}$, $\frac{1}{45} a^{16} + \frac{2}{15} a^{14} - \frac{2}{15} a^{13} + \frac{1}{5} a^{12} + \frac{19}{45} a^{11} + \frac{17}{45} a^{10} + \frac{2}{15} a^{9} + \frac{16}{45} a^{8} - \frac{19}{45} a^{7} + \frac{4}{9} a^{6} + \frac{1}{9} a^{5} - \frac{17}{45} a^{4} + \frac{7}{15} a^{3} + \frac{22}{45} a^{2} + \frac{4}{15} a$, $\frac{1}{275364213347514735} a^{17} + \frac{192069238747606}{30596023705279415} a^{16} - \frac{2358658481336029}{91788071115838245} a^{15} + \frac{4366036721060311}{91788071115838245} a^{14} - \frac{14580778044201687}{30596023705279415} a^{13} + \frac{2778211656717418}{275364213347514735} a^{12} + \frac{112863435007184261}{275364213347514735} a^{11} + \frac{5928878695758299}{91788071115838245} a^{10} - \frac{16020149815768556}{275364213347514735} a^{9} + \frac{63377835809834582}{275364213347514735} a^{8} + \frac{134735604253009616}{275364213347514735} a^{7} - \frac{123728738531045728}{275364213347514735} a^{6} + \frac{136074546720408538}{275364213347514735} a^{5} + \frac{43252579338235756}{91788071115838245} a^{4} - \frac{24279131928709868}{275364213347514735} a^{3} + \frac{10281395541507214}{91788071115838245} a^{2} - \frac{5903954616159947}{30596023705279415} a + \frac{10540375619730916}{30596023705279415}$

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:  $17$
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:  \( 866588373.731 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3\times A_4$ (as 18T60):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 24 conjugacy class representatives for $C_2\times S_3\times A_4$
Character table for $C_2\times S_3\times A_4$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.469.1, 6.6.434581.1, 9.9.247691263309.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$7.6.4.3$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}$
$67$67.3.0.1$x^{3} - x + 16$$1$$3$$0$$C_3$$[\ ]^{3}$
67.3.0.1$x^{3} - x + 16$$1$$3$$0$$C_3$$[\ ]^{3}$
67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
181Data not computed