Properties

Label 18.0.16782375028...6571.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,139\cdot 367^{2}\cdot 299401^{2}$
Root discriminant $10.29$
Ramified primes $139, 367, 299401$
Class number $1$
Class group Trivial
Galois Group 18T968

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 29, -60, 74, -44, -16, 56, -48, 9, 24, -29, 13, 5, -10, 5, 1, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + x^16 + 5*x^15 - 10*x^14 + 5*x^13 + 13*x^12 - 29*x^11 + 24*x^10 + 9*x^9 - 48*x^8 + 56*x^7 - 16*x^6 - 44*x^5 + 74*x^4 - 60*x^3 + 29*x^2 - 8*x + 1)
gp: K = bnfinit(x^18 - 2*x^17 + x^16 + 5*x^15 - 10*x^14 + 5*x^13 + 13*x^12 - 29*x^11 + 24*x^10 + 9*x^9 - 48*x^8 + 56*x^7 - 16*x^6 - 44*x^5 + 74*x^4 - 60*x^3 + 29*x^2 - 8*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 2 x^{17} \) \(\mathstrut +\mathstrut x^{16} \) \(\mathstrut +\mathstrut 5 x^{15} \) \(\mathstrut -\mathstrut 10 x^{14} \) \(\mathstrut +\mathstrut 5 x^{13} \) \(\mathstrut +\mathstrut 13 x^{12} \) \(\mathstrut -\mathstrut 29 x^{11} \) \(\mathstrut +\mathstrut 24 x^{10} \) \(\mathstrut +\mathstrut 9 x^{9} \) \(\mathstrut -\mathstrut 48 x^{8} \) \(\mathstrut +\mathstrut 56 x^{7} \) \(\mathstrut -\mathstrut 16 x^{6} \) \(\mathstrut -\mathstrut 44 x^{5} \) \(\mathstrut +\mathstrut 74 x^{4} \) \(\mathstrut -\mathstrut 60 x^{3} \) \(\mathstrut +\mathstrut 29 x^{2} \) \(\mathstrut -\mathstrut 8 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-1678237502892756571=-\,139\cdot 367^{2}\cdot 299401^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.29$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $139, 367, 299401$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{15073} a^{17} + \frac{1035}{15073} a^{16} + \frac{3113}{15073} a^{15} + \frac{2564}{15073} a^{14} + \frac{6010}{15073} a^{13} + \frac{7226}{15073} a^{12} + \frac{2094}{15073} a^{11} + \frac{937}{15073} a^{10} + \frac{7021}{15073} a^{9} + \frac{527}{15073} a^{8} + \frac{3823}{15073} a^{7} + \frac{308}{15073} a^{6} + \frac{2847}{15073} a^{5} - \frac{2013}{15073} a^{4} - \frac{7333}{15073} a^{3} + \frac{7484}{15073} a^{2} - \frac{1658}{15073} a - \frac{1032}{15073}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( \frac{28906}{15073} a^{17} - \frac{47414}{15073} a^{16} + \frac{28714}{15073} a^{15} + \frac{136700}{15073} a^{14} - \frac{247506}{15073} a^{13} + \frac{143852}{15073} a^{12} + \frac{342675}{15073} a^{11} - \frac{754909}{15073} a^{10} + \frac{654293}{15073} a^{9} + \frac{235827}{15073} a^{8} - \frac{1258657}{15073} a^{7} + \frac{1502205}{15073} a^{6} - \frac{455388}{15073} a^{5} - \frac{1151546}{15073} a^{4} + \frac{2008610}{15073} a^{3} - \frac{1668295}{15073} a^{2} + \frac{744569}{15073} a - \frac{152255}{15073} \),  \( \frac{110167}{15073} a^{17} - \frac{155130}{15073} a^{16} + \frac{8975}{15073} a^{15} + \frac{572942}{15073} a^{14} - \frac{761651}{15073} a^{13} + \frac{46539}{15073} a^{12} + \frac{1534879}{15073} a^{11} - \frac{2284521}{15073} a^{10} + \frac{1141987}{15073} a^{9} + \frac{1880938}{15073} a^{8} - \frac{4266984}{15073} a^{7} + \frac{3453830}{15073} a^{6} + \frac{684750}{15073} a^{5} - \frac{4745117}{15073} a^{4} + \frac{5243301}{15073} a^{3} - \frac{3048018}{15073} a^{2} + \frac{917181}{15073} a - \frac{102216}{15073} \),  \( a \),  \( \frac{62425}{15073} a^{17} - \frac{68368}{15073} a^{16} - \frac{7164}{15073} a^{15} + \frac{314046}{15073} a^{14} - \frac{339399}{15073} a^{13} - \frac{36767}{15073} a^{12} + \frac{818836}{15073} a^{11} - \frac{1061198}{15073} a^{10} + \frac{430348}{15073} a^{9} + \frac{1078872}{15073} a^{8} - \frac{2065035}{15073} a^{7} + \frac{1485979}{15073} a^{6} + \frac{601152}{15073} a^{5} - \frac{2379458}{15073} a^{4} + \frac{2370946}{15073} a^{3} - \frac{1295213}{15073} a^{2} + \frac{352320}{15073} a - \frac{30744}{15073} \),  \( \frac{109774}{15073} a^{17} - \frac{124768}{15073} a^{16} - \frac{23667}{15073} a^{15} + \frac{560108}{15073} a^{14} - \frac{606390}{15073} a^{13} - \frac{110285}{15073} a^{12} + \frac{1480660}{15073} a^{11} - \frac{1868966}{15073} a^{10} + \frac{658757}{15073} a^{9} + \frac{2005433}{15073} a^{8} - \frac{3674276}{15073} a^{7} + \frac{2473625}{15073} a^{6} + \frac{1269128}{15073} a^{5} - \frac{4300687}{15073} a^{4} + \frac{4055460}{15073} a^{3} - \frac{2025031}{15073} a^{2} + \frac{483519}{15073} a - \frac{28246}{15073} \),  \( \frac{109533}{15073} a^{17} - \frac{117962}{15073} a^{16} - \frac{5177}{15073} a^{15} + \frac{545104}{15073} a^{14} - \frac{592719}{15073} a^{13} - \frac{27918}{15073} a^{12} + \frac{1413123}{15073} a^{11} - \frac{1868688}{15073} a^{10} + \frac{835748}{15073} a^{9} + \frac{1818134}{15073} a^{8} - \frac{3600801}{15073} a^{7} + \frac{2715930}{15073} a^{6} + \frac{884461}{15073} a^{5} - \frac{4086868}{15073} a^{4} + \frac{4270194}{15073} a^{3} - \frac{2472105}{15073} a^{2} + \frac{762513}{15073} a - \frac{96067}{15073} \),  \( \frac{43873}{15073} a^{17} - \frac{96832}{15073} a^{16} + \frac{30342}{15073} a^{15} + \frac{241741}{15073} a^{14} - \frac{477595}{15073} a^{13} + \frac{146619}{15073} a^{12} + \frac{678412}{15073} a^{11} - \frac{1351640}{15073} a^{10} + \frac{889812}{15073} a^{9} + \frac{662301}{15073} a^{8} - \frac{2342180}{15073} a^{7} + \frac{2298572}{15073} a^{6} - \frac{229615}{15073} a^{5} - \frac{2415322}{15073} a^{4} + \frac{3223025}{15073} a^{3} - \frac{2129993}{15073} a^{2} + \frac{769587}{15073} a - \frac{118228}{15073} \),  \( \frac{25055}{15073} a^{17} - \frac{38854}{15073} a^{16} - \frac{6560}{15073} a^{15} + \frac{135551}{15073} a^{14} - \frac{179596}{15073} a^{13} - \frac{24519}{15073} a^{12} + \frac{372882}{15073} a^{11} - \frac{519681}{15073} a^{10} + \frac{190121}{15073} a^{9} + \frac{497446}{15073} a^{8} - \frac{998468}{15073} a^{7} + \frac{692922}{15073} a^{6} + \frac{307609}{15073} a^{5} - \frac{1147005}{15073} a^{4} + \frac{1111884}{15073} a^{3} - \frac{524055}{15073} a^{2} + \frac{120582}{15073} a + \frac{8508}{15073} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 27.3003078572 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

18T968:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 185794560
The 300 conjugacy class representatives for t18n968 are not computed
Character table for t18n968 is not computed

Intermediate fields

9.3.109880167.1

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 $18$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ $18$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
139Data not computed
367Data not computed
299401Data not computed