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Properties

Label 18.0.25066383071...8467.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 1399^{2}\cdot 4523^{3}$
Root discriminant $33.28$
Ramified primes $7, 1399, 4523$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group 18T472

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![912673, -639812, -43068, -28704, 215570, -83059, 36052, -9658, -10699, 6918, -2627, 1255, -41, -100, 58, -38, 13, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 13*x^16 - 38*x^15 + 58*x^14 - 100*x^13 - 41*x^12 + 1255*x^11 - 2627*x^10 + 6918*x^9 - 10699*x^8 - 9658*x^7 + 36052*x^6 - 83059*x^5 + 215570*x^4 - 28704*x^3 - 43068*x^2 - 639812*x + 912673)
 
gp: K = bnfinit(x^18 - 3*x^17 + 13*x^16 - 38*x^15 + 58*x^14 - 100*x^13 - 41*x^12 + 1255*x^11 - 2627*x^10 + 6918*x^9 - 10699*x^8 - 9658*x^7 + 36052*x^6 - 83059*x^5 + 215570*x^4 - 28704*x^3 - 43068*x^2 - 639812*x + 912673, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 13 x^{16} - 38 x^{15} + 58 x^{14} - 100 x^{13} - 41 x^{12} + 1255 x^{11} - 2627 x^{10} + 6918 x^{9} - 10699 x^{8} - 9658 x^{7} + 36052 x^{6} - 83059 x^{5} + 215570 x^{4} - 28704 x^{3} - 43068 x^{2} - 639812 x + 912673 \)

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:  \(-2506638307140843254915898467=-\,7^{12}\cdot 1399^{2}\cdot 4523^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.28$
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:  $7, 1399, 4523$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{29} a^{15} + \frac{10}{29} a^{14} + \frac{1}{29} a^{13} - \frac{6}{29} a^{12} - \frac{11}{29} a^{11} - \frac{11}{29} a^{10} - \frac{6}{29} a^{9} - \frac{11}{29} a^{8} + \frac{1}{29} a^{7} + \frac{4}{29} a^{6} + \frac{4}{29} a^{5} - \frac{6}{29} a^{4} + \frac{11}{29} a^{3} - \frac{9}{29} a^{2} - \frac{5}{29} a - \frac{4}{29}$, $\frac{1}{2813} a^{16} - \frac{3}{2813} a^{15} + \frac{886}{2813} a^{14} - \frac{135}{2813} a^{13} - \frac{136}{2813} a^{12} - \frac{100}{2813} a^{11} + \frac{1123}{2813} a^{10} - \frac{1267}{2813} a^{9} - \frac{784}{2813} a^{8} - \frac{357}{2813} a^{7} + \frac{68}{2813} a^{6} - \frac{32}{97} a^{5} - \frac{1390}{2813} a^{4} + \frac{167}{2813} a^{3} - \frac{352}{2813} a^{2} + \frac{1366}{2813} a + \frac{11}{29}$, $\frac{1}{3308203377705608178254846514553285563923148827} a^{17} - \frac{374902716252469597336194929779586697486789}{3308203377705608178254846514553285563923148827} a^{16} + \frac{1064861667770996757380123478020196106501668}{3308203377705608178254846514553285563923148827} a^{15} - \frac{480943993296867248777379639785156938896252282}{3308203377705608178254846514553285563923148827} a^{14} - \frac{1197020849733927642973923766005543902099984982}{3308203377705608178254846514553285563923148827} a^{13} + \frac{1268169623423217731165915218539297851247703226}{3308203377705608178254846514553285563923148827} a^{12} + \frac{462769491207785713785164146619520124433458739}{3308203377705608178254846514553285563923148827} a^{11} + \frac{1027511569818637256534431418793319801311334390}{3308203377705608178254846514553285563923148827} a^{10} - \frac{63930701913965836261187686011602551659655014}{254477182900431398327295885734868120301780679} a^{9} - \frac{1033830660462985567908691872204382433359638626}{3308203377705608178254846514553285563923148827} a^{8} + \frac{250360914022466518438834718779312799807395688}{3308203377705608178254846514553285563923148827} a^{7} + \frac{326425396738722649368451824636723386775138709}{3308203377705608178254846514553285563923148827} a^{6} + \frac{1071845371984875272778840730572440242247987756}{3308203377705608178254846514553285563923148827} a^{5} - \frac{1601744599726963996618838499549111357815100478}{3308203377705608178254846514553285563923148827} a^{4} - \frac{613648271249907548291805641786939048707279507}{3308203377705608178254846514553285563923148827} a^{3} + \frac{381011772556836966142134333620114213956679433}{3308203377705608178254846514553285563923148827} a^{2} - \frac{50299374657665933461639786364751095787802}{1176041015892502018576198547654918437228279} a - \frac{22103436912090958546266741604915249941904}{351599891349304727203193380226728192573403}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 216238.286744 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T472:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5184
The 88 conjugacy class representatives for t18n472 are not computed
Character table for t18n472 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.0.10859723.1, 9.3.164590951.1

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 $18$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
7Data not computed
1399Data not computed
4523Data not computed