Properties

Label 18.0.92919321348...9543.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{15}\cdot 1399^{2}$
Root discriminant $11.32$
Ramified primes $7, 1399$
Class number $1$
Class group Trivial
Galois Group 18T286

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 11, -20, 30, -39, 52, -71, 92, -106, 107, -97, 80, -58, 37, -21, 10, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 10*x^16 - 21*x^15 + 37*x^14 - 58*x^13 + 80*x^12 - 97*x^11 + 107*x^10 - 106*x^9 + 92*x^8 - 71*x^7 + 52*x^6 - 39*x^5 + 30*x^4 - 20*x^3 + 11*x^2 - 4*x + 1)
gp: K = bnfinit(x^18 - 4*x^17 + 10*x^16 - 21*x^15 + 37*x^14 - 58*x^13 + 80*x^12 - 97*x^11 + 107*x^10 - 106*x^9 + 92*x^8 - 71*x^7 + 52*x^6 - 39*x^5 + 30*x^4 - 20*x^3 + 11*x^2 - 4*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 4 x^{17} \) \(\mathstrut +\mathstrut 10 x^{16} \) \(\mathstrut -\mathstrut 21 x^{15} \) \(\mathstrut +\mathstrut 37 x^{14} \) \(\mathstrut -\mathstrut 58 x^{13} \) \(\mathstrut +\mathstrut 80 x^{12} \) \(\mathstrut -\mathstrut 97 x^{11} \) \(\mathstrut +\mathstrut 107 x^{10} \) \(\mathstrut -\mathstrut 106 x^{9} \) \(\mathstrut +\mathstrut 92 x^{8} \) \(\mathstrut -\mathstrut 71 x^{7} \) \(\mathstrut +\mathstrut 52 x^{6} \) \(\mathstrut -\mathstrut 39 x^{5} \) \(\mathstrut +\mathstrut 30 x^{4} \) \(\mathstrut -\mathstrut 20 x^{3} \) \(\mathstrut +\mathstrut 11 x^{2} \) \(\mathstrut -\mathstrut 4 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-9291932134821949543=-\,7^{15}\cdot 1399^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.32$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $7, 1399$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{841} a^{17} - \frac{92}{841} a^{16} - \frac{304}{841} a^{15} - \frac{181}{841} a^{14} - \frac{14}{841} a^{13} + \frac{333}{841} a^{12} + \frac{211}{841} a^{11} - \frac{163}{841} a^{10} + \frac{154}{841} a^{9} - \frac{202}{841} a^{8} + \frac{207}{841} a^{7} + \frac{215}{841} a^{6} - \frac{366}{841} a^{5} + \frac{211}{841} a^{4} - \frac{36}{841} a^{3} - \frac{216}{841} a^{2} - \frac{324}{841} a - \frac{86}{841}$

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:  \( -\frac{1834}{841} a^{17} + \frac{5574}{841} a^{16} - \frac{11821}{841} a^{15} + \frac{24148}{841} a^{14} - \frac{39081}{841} a^{13} + \frac{57873}{841} a^{12} - \frac{74963}{841} a^{11} + \frac{83646}{841} a^{10} - \frac{89847}{841} a^{9} + \frac{82846}{841} a^{8} - \frac{63422}{841} a^{7} + \frac{48897}{841} a^{6} - \frac{37719}{841} a^{5} + \frac{26798}{841} a^{4} - \frac{19758}{841} a^{3} + \frac{10966}{841} a^{2} - \frac{5417}{841} a + \frac{1298}{841} \) (order $14$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{1762}{841} a^{17} - \frac{6519}{841} a^{16} + \frac{14366}{841} a^{15} - \frac{28777}{841} a^{14} + \frac{48499}{841} a^{13} - \frac{71757}{841} a^{12} + \frac{94252}{841} a^{11} - \frac{106391}{841} a^{10} + \frac{111558}{841} a^{9} - \frac{105306}{841} a^{8} + \frac{81317}{841} a^{7} - \frac{57649}{841} a^{6} + \frac{44728}{841} a^{5} - \frac{35262}{841} a^{4} + \frac{25714}{841} a^{3} - \frac{14757}{841} a^{2} + \frac{6038}{841} a - \frac{1834}{841} \),  \( \frac{1816}{841} a^{17} - \frac{7282}{841} a^{16} + \frac{16452}{841} a^{15} - \frac{33505}{841} a^{14} + \frac{57835}{841} a^{13} - \frac{86574}{841} a^{12} + \frac{116579}{841} a^{11} - \frac{134536}{841} a^{10} + \frac{143422}{841} a^{9} - \frac{138921}{841} a^{8} + \frac{110997}{841} a^{7} - \frac{81361}{841} a^{6} + \frac{61968}{841} a^{5} - \frac{46575}{841} a^{4} + \frac{34703}{841} a^{3} - \frac{22216}{841} a^{2} + \frac{9567}{841} a - \frac{3955}{841} \),  \( \frac{2529}{841} a^{17} - \frac{8121}{841} a^{16} + \frac{16678}{841} a^{15} - \frac{32203}{841} a^{14} + \frac{51217}{841} a^{13} - \frac{72851}{841} a^{12} + \frac{89571}{841} a^{11} - \frac{93488}{841} a^{10} + \frac{92593}{841} a^{9} - \frac{79425}{841} a^{8} + \frac{50861}{841} a^{7} - \frac{29827}{841} a^{6} + \frac{25557}{841} a^{5} - \frac{22282}{841} a^{4} + \frac{14081}{841} a^{3} - \frac{2978}{841} a^{2} - \frac{1103}{841} a + \frac{2007}{841} \),  \( \frac{2836}{841} a^{17} - \frac{11135}{841} a^{16} + \frac{25952}{841} a^{15} - \frac{52448}{841} a^{14} + \frac{89810}{841} a^{13} - \frac{135456}{841} a^{12} + \frac{180419}{841} a^{11} - \frac{208286}{841} a^{10} + \frac{219766}{841} a^{9} - \frac{209560}{841} a^{8} + \frac{168234}{841} a^{7} - \frac{119407}{841} a^{6} + \frac{88123}{841} a^{5} - \frac{69358}{841} a^{4} + \frac{54330}{841} a^{3} - \frac{32286}{841} a^{2} + \frac{12964}{841} a - \frac{3370}{841} \),  \( \frac{1591}{841} a^{17} - \frac{4243}{841} a^{16} + \frac{8321}{841} a^{15} - \frac{15487}{841} a^{14} + \frac{22299}{841} a^{13} - \frac{30303}{841} a^{12} + \frac{32941}{841} a^{11} - \frac{28899}{841} a^{10} + \frac{24672}{841} a^{9} - \frac{13576}{841} a^{8} - \frac{335}{841} a^{7} + \frac{5665}{841} a^{6} - \frac{3698}{841} a^{5} - \frac{699}{841} a^{4} + \frac{753}{841} a^{3} + \frac{5359}{841} a^{2} - \frac{4997}{841} a + \frac{3621}{841} \),  \( \frac{2762}{841} a^{17} - \frac{9373}{841} a^{16} + \frac{20695}{841} a^{15} - \frac{40736}{841} a^{14} + \frac{66457}{841} a^{13} - \frac{97864}{841} a^{12} + \frac{124437}{841} a^{11} - \frac{136513}{841} a^{10} + \frac{139408}{841} a^{9} - \frac{124809}{841} a^{8} + \frac{91523}{841} a^{7} - \frac{59627}{841} a^{6} + \frac{43722}{841} a^{5} - \frac{37876}{841} a^{4} + \frac{29241}{841} a^{3} - \frac{11256}{841} a^{2} + \frac{2459}{841} a + \frac{1312}{841} \),  \( \frac{1596}{841} a^{17} - \frac{2180}{841} a^{16} + \frac{914}{841} a^{15} + \frac{428}{841} a^{14} - \frac{8888}{841} a^{13} + \frac{20981}{841} a^{12} - \frac{41694}{841} a^{11} + \frac{66160}{841} a^{10} - \frac{78842}{841} a^{9} + \frac{91380}{841} a^{8} - \frac{95174}{841} a^{7} + \frac{74020}{841} a^{6} - \frac{46737}{841} a^{5} + \frac{33996}{841} a^{4} - \frac{28862}{841} a^{3} + \frac{26986}{841} a^{2} - \frac{14186}{841} a + \frac{5714}{841} \),  \( \frac{1919}{841} a^{17} - \frac{8348}{841} a^{16} + \frac{20462}{841} a^{15} - \frac{42056}{841} a^{14} + \frac{74054}{841} a^{13} - \frac{113668}{841} a^{12} + \frac{155132}{841} a^{11} - \frac{184124}{841} a^{10} + \frac{197129}{841} a^{9} - \frac{192526}{841} a^{8} + \frac{160071}{841} a^{7} - \frac{116404}{841} a^{6} + \frac{83981}{841} a^{5} - \frac{63528}{841} a^{4} + \frac{50338}{841} a^{3} - \frac{32690}{841} a^{2} + \frac{14040}{841} a - \frac{3562}{841} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 509.337225135 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

18T286:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 1296
The 34 conjugacy class representatives for t18n286
Character table for t18n286 is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $18$

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
7Data not computed
1399Data not computed