Properties

Label 18.0.31908518100...3743.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,11^{6}\cdot 23^{9}$
Root discriminant $10.67$
Ramified primes $11, 23$
Class number $1$
Class group Trivial
Galois Group 18T221

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 15, -36, 75, -137, 221, -312, 384, -411, 384, -312, 221, -137, 75, -36, 15, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 15*x^16 - 36*x^15 + 75*x^14 - 137*x^13 + 221*x^12 - 312*x^11 + 384*x^10 - 411*x^9 + 384*x^8 - 312*x^7 + 221*x^6 - 137*x^5 + 75*x^4 - 36*x^3 + 15*x^2 - 5*x + 1)
gp: K = bnfinit(x^18 - 5*x^17 + 15*x^16 - 36*x^15 + 75*x^14 - 137*x^13 + 221*x^12 - 312*x^11 + 384*x^10 - 411*x^9 + 384*x^8 - 312*x^7 + 221*x^6 - 137*x^5 + 75*x^4 - 36*x^3 + 15*x^2 - 5*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 5 x^{17} \) \(\mathstrut +\mathstrut 15 x^{16} \) \(\mathstrut -\mathstrut 36 x^{15} \) \(\mathstrut +\mathstrut 75 x^{14} \) \(\mathstrut -\mathstrut 137 x^{13} \) \(\mathstrut +\mathstrut 221 x^{12} \) \(\mathstrut -\mathstrut 312 x^{11} \) \(\mathstrut +\mathstrut 384 x^{10} \) \(\mathstrut -\mathstrut 411 x^{9} \) \(\mathstrut +\mathstrut 384 x^{8} \) \(\mathstrut -\mathstrut 312 x^{7} \) \(\mathstrut +\mathstrut 221 x^{6} \) \(\mathstrut -\mathstrut 137 x^{5} \) \(\mathstrut +\mathstrut 75 x^{4} \) \(\mathstrut -\mathstrut 36 x^{3} \) \(\mathstrut +\mathstrut 15 x^{2} \) \(\mathstrut -\mathstrut 5 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-3190851810094053743=-\,11^{6}\cdot 23^{9}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.67$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{11} a^{16} + \frac{4}{11} a^{15} - \frac{5}{11} a^{14} + \frac{3}{11} a^{13} - \frac{3}{11} a^{12} - \frac{2}{11} a^{11} - \frac{3}{11} a^{10} + \frac{4}{11} a^{9} + \frac{5}{11} a^{8} + \frac{4}{11} a^{7} - \frac{3}{11} a^{6} - \frac{2}{11} a^{5} - \frac{3}{11} a^{4} + \frac{3}{11} a^{3} - \frac{5}{11} a^{2} + \frac{4}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{17} + \frac{1}{11} a^{15} + \frac{1}{11} a^{14} - \frac{4}{11} a^{13} - \frac{1}{11} a^{12} + \frac{5}{11} a^{11} + \frac{5}{11} a^{10} - \frac{5}{11} a^{8} + \frac{3}{11} a^{7} - \frac{1}{11} a^{6} + \frac{5}{11} a^{5} + \frac{4}{11} a^{4} + \frac{5}{11} a^{3} + \frac{2}{11} a^{2} - \frac{4}{11} a - \frac{4}{11}$

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:  \( \frac{23}{11} a^{17} - \frac{109}{11} a^{16} + \frac{302}{11} a^{15} - \frac{686}{11} a^{14} + \frac{1385}{11} a^{13} - \frac{2435}{11} a^{12} + \frac{3754}{11} a^{11} - \frac{5025}{11} a^{10} + \frac{5790}{11} a^{9} - 523 a^{8} + \frac{4990}{11} a^{7} - \frac{3733}{11} a^{6} + \frac{2434}{11} a^{5} - \frac{1407}{11} a^{4} + \frac{723}{11} a^{3} - \frac{300}{11} a^{2} + 10 a - \frac{25}{11} \),  \( \frac{56}{11} a^{17} - \frac{255}{11} a^{16} + \frac{697}{11} a^{15} - 145 a^{14} + \frac{3202}{11} a^{13} - \frac{5605}{11} a^{12} + \frac{8633}{11} a^{11} - 1047 a^{10} + \frac{13225}{11} a^{9} - \frac{13116}{11} a^{8} + \frac{11259}{11} a^{7} - \frac{8333}{11} a^{6} + \frac{5333}{11} a^{5} - \frac{3037}{11} a^{4} + \frac{1517}{11} a^{3} - \frac{670}{11} a^{2} + \frac{241}{11} a - \frac{61}{11} \),  \( \frac{5}{11} a^{17} - \frac{28}{11} a^{16} + \frac{80}{11} a^{15} - \frac{196}{11} a^{14} + \frac{413}{11} a^{13} - \frac{757}{11} a^{12} + \frac{1225}{11} a^{11} - \frac{1750}{11} a^{10} + \frac{2165}{11} a^{9} - 214 a^{8} + \frac{2213}{11} a^{7} - \frac{1802}{11} a^{6} + \frac{1236}{11} a^{5} - \frac{754}{11} a^{4} + \frac{370}{11} a^{3} - \frac{169}{11} a^{2} + 6 a - \frac{26}{11} \),  \( \frac{2}{11} a^{17} + \frac{3}{11} a^{16} - \frac{30}{11} a^{15} + \frac{97}{11} a^{14} - \frac{230}{11} a^{13} + 44 a^{12} - \frac{887}{11} a^{11} + \frac{1409}{11} a^{10} - \frac{1913}{11} a^{9} + \frac{2216}{11} a^{8} - \frac{2160}{11} a^{7} + 166 a^{6} - \frac{1316}{11} a^{5} + \frac{813}{11} a^{4} - \frac{432}{11} a^{3} + 19 a^{2} - \frac{73}{11} a + \frac{17}{11} \),  \( \frac{15}{11} a^{17} - \frac{39}{11} a^{16} + \frac{90}{11} a^{15} - \frac{186}{11} a^{14} + \frac{329}{11} a^{13} - \frac{492}{11} a^{12} + \frac{648}{11} a^{11} - \frac{655}{11} a^{10} + \frac{548}{11} a^{9} - \frac{303}{11} a^{8} + \frac{87}{11} a^{7} + \frac{102}{11} a^{6} - \frac{111}{11} a^{5} + \frac{111}{11} a^{4} - \frac{53}{11} a^{3} + \frac{27}{11} a^{2} - \frac{7}{11} a \),  \( \frac{5}{11} a^{17} - \frac{42}{11} a^{16} + \frac{123}{11} a^{15} - \frac{302}{11} a^{14} + \frac{635}{11} a^{13} - 106 a^{12} + \frac{1880}{11} a^{11} - \frac{2665}{11} a^{10} + \frac{3209}{11} a^{9} - \frac{3381}{11} a^{8} + \frac{3026}{11} a^{7} - 216 a^{6} + \frac{1561}{11} a^{5} - \frac{943}{11} a^{4} + \frac{460}{11} a^{3} - 20 a^{2} + \frac{76}{11} a - \frac{29}{11} \),  \( \frac{31}{11} a^{17} - \frac{112}{11} a^{16} + \frac{276}{11} a^{15} - \frac{597}{11} a^{14} + \frac{1135}{11} a^{13} - \frac{1862}{11} a^{12} + \frac{2678}{11} a^{11} - \frac{3238}{11} a^{10} + \frac{3325}{11} a^{9} - 261 a^{8} + \frac{2120}{11} a^{7} - \frac{1290}{11} a^{6} + \frac{698}{11} a^{5} - \frac{354}{11} a^{4} + \frac{171}{11} a^{3} - \frac{71}{11} a^{2} + 2 a + \frac{6}{11} \),  \( \frac{30}{11} a^{17} - \frac{136}{11} a^{16} + \frac{388}{11} a^{15} - \frac{907}{11} a^{14} + 168 a^{13} - \frac{3296}{11} a^{12} + \frac{5185}{11} a^{11} - \frac{7098}{11} a^{10} + \frac{8443}{11} a^{9} - \frac{8684}{11} a^{8} + \frac{7763}{11} a^{7} - \frac{5980}{11} a^{6} + \frac{3986}{11} a^{5} - 210 a^{4} + \frac{1161}{11} a^{3} - \frac{503}{11} a^{2} + \frac{183}{11} a - \frac{47}{11} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 39.4992656122 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

18T221:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 648
The 14 conjugacy class representatives for t18n221
Character table for t18n221

Intermediate fields

3.1.23.1, 6.0.1472207.1, 9.1.33860761.1 x3

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

Sibling fields

Degree 9 sibling: data not computed
Degree 12 siblings: data not computed
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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$11$11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$