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Properties

Label 18.0.43529619286...4559.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 7^{9}\cdot 19^{17}$
Root discriminant $73.93$
Ramified primes $3, 7, 19$
Class number $53056$ (GRH)
Class group $[2, 2, 2, 6632]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![295252021, -258142646, 258142646, -146814521, 146814521, -48845771, 48845771, -9658271, 9658271, -1167646, 1167646, -87021, 87021, -3896, 3896, -96, 96, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 96*x^16 - 96*x^15 + 3896*x^14 - 3896*x^13 + 87021*x^12 - 87021*x^11 + 1167646*x^10 - 1167646*x^9 + 9658271*x^8 - 9658271*x^7 + 48845771*x^6 - 48845771*x^5 + 146814521*x^4 - 146814521*x^3 + 258142646*x^2 - 258142646*x + 295252021)
 
gp: K = bnfinit(x^18 - x^17 + 96*x^16 - 96*x^15 + 3896*x^14 - 3896*x^13 + 87021*x^12 - 87021*x^11 + 1167646*x^10 - 1167646*x^9 + 9658271*x^8 - 9658271*x^7 + 48845771*x^6 - 48845771*x^5 + 146814521*x^4 - 146814521*x^3 + 258142646*x^2 - 258142646*x + 295252021, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 96 x^{16} - 96 x^{15} + 3896 x^{14} - 3896 x^{13} + 87021 x^{12} - 87021 x^{11} + 1167646 x^{10} - 1167646 x^{9} + 9658271 x^{8} - 9658271 x^{7} + 48845771 x^{6} - 48845771 x^{5} + 146814521 x^{4} - 146814521 x^{3} + 258142646 x^{2} - 258142646 x + 295252021 \)

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:  \(-4352961928683212902721499543224559=-\,3^{9}\cdot 7^{9}\cdot 19^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.93$
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:  $3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(399=3\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{399}(64,·)$, $\chi_{399}(1,·)$, $\chi_{399}(230,·)$, $\chi_{399}(398,·)$, $\chi_{399}(335,·)$, $\chi_{399}(146,·)$, $\chi_{399}(85,·)$, $\chi_{399}(356,·)$, $\chi_{399}(293,·)$, $\chi_{399}(358,·)$, $\chi_{399}(167,·)$, $\chi_{399}(232,·)$, $\chi_{399}(41,·)$, $\chi_{399}(106,·)$, $\chi_{399}(43,·)$, $\chi_{399}(169,·)$, $\chi_{399}(314,·)$, $\chi_{399}(253,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{64457461} a^{10} + \frac{10630121}{64457461} a^{9} + \frac{50}{64457461} a^{8} + \frac{27153218}{64457461} a^{7} + \frac{875}{64457461} a^{6} + \frac{20553504}{64457461} a^{5} + \frac{6250}{64457461} a^{4} + \frac{28242852}{64457461} a^{3} + \frac{15625}{64457461} a^{2} - \frac{22093183}{64457461} a + \frac{6250}{64457461}$, $\frac{1}{64457461} a^{11} + \frac{55}{64457461} a^{9} + \frac{11306856}{64457461} a^{8} + \frac{1100}{64457461} a^{7} + \frac{1072013}{64457461} a^{6} + \frac{9625}{64457461} a^{5} - \frac{18828568}{64457461} a^{4} + \frac{34375}{64457461} a^{3} - \frac{10856811}{64457461} a^{2} + \frac{34375}{64457461} a + \frac{17386041}{64457461}$, $\frac{1}{64457461} a^{12} + \frac{6767350}{64457461} a^{9} - \frac{1650}{64457461} a^{8} - \frac{9833374}{64457461} a^{7} - \frac{38500}{64457461} a^{6} + \frac{10963010}{64457461} a^{5} - \frac{309375}{64457461} a^{4} - \frac{17234607}{64457461} a^{3} - \frac{825000}{64457461} a^{2} + \frac{7819347}{64457461} a - \frac{343750}{64457461}$, $\frac{1}{64457461} a^{13} - \frac{1950}{64457461} a^{9} - \frac{25913569}{64457461} a^{8} - \frac{52000}{64457461} a^{7} + \frac{19618172}{64457461} a^{6} - \frac{511875}{64457461} a^{5} - \frac{29077691}{64457461} a^{4} - \frac{1950000}{64457461} a^{3} - \frac{21788363}{64457461} a^{2} - \frac{2031250}{64457461} a - \frac{11843084}{64457461}$, $\frac{1}{64457461} a^{14} + \frac{11977400}{64457461} a^{9} + \frac{45500}{64457461} a^{8} - \frac{15639670}{64457461} a^{7} + \frac{1194375}{64457461} a^{6} + \frac{22171828}{64457461} a^{5} + \frac{10237500}{64457461} a^{4} + \frac{5101343}{64457461} a^{3} + \frac{28437500}{64457461} a^{2} + \frac{28491475}{64457461} a + \frac{12187500}{64457461}$, $\frac{1}{64457461} a^{15} + \frac{56875}{64457461} a^{9} + \frac{30064940}{64457461} a^{8} + \frac{1706250}{64457461} a^{7} - \frac{15944490}{64457461} a^{6} + \frac{17915625}{64457461} a^{5} - \frac{18536436}{64457461} a^{4} + \frac{6636289}{64457461} a^{3} + \frac{1625758}{64457461} a^{2} + \frac{11714414}{64457461} a - \frac{23637779}{64457461}$, $\frac{1}{64457461} a^{16} - \frac{11540216}{64457461} a^{9} - \frac{1137500}{64457461} a^{8} - \frac{18910141}{64457461} a^{7} - \frac{31850000}{64457461} a^{6} + \frac{1436260}{64457461} a^{5} - \frac{26545156}{64457461} a^{4} - \frac{30653622}{64457461} a^{3} + \frac{25446993}{64457461} a^{2} - \frac{7599388}{64457461} a + \frac{31276016}{64457461}$, $\frac{1}{64457461} a^{17} - \frac{1487500}{64457461} a^{9} - \frac{22016490}{64457461} a^{8} + \frac{16857461}{64457461} a^{7} - \frac{20696117}{64457461} a^{6} - \frac{4965312}{64457461} a^{5} - \frac{32202481}{64457461} a^{4} + \frac{2096213}{64457461} a^{3} + \frac{20757195}{64457461} a^{2} - \frac{3750154}{64457461} a - \frac{1548859}{64457461}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6632}$, which has order $53056$ (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:  \( -1 \) (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:  \( 22305.8950792 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-399}) \), 3.3.361.1, 6.0.22931152839.1, \(\Q(\zeta_{19})^+\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
3Data not computed
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19Data not computed