# Properties

 Label 18.0.11126075312...9771.1 Degree $18$ Signature $[0, 9]$ Discriminant $-\,3^{4}\cdot 11^{5}\cdot 31^{8}$ Root discriminant $11.43$ Ramified primes $3, 11, 31$ Class number $1$ Class group Trivial Galois Group 18T485

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 7, 3, -14, 15, -6, -16, 44, -37, -19, 69, -55, -8, 55, -54, 28, -8, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 28*x^16 - 54*x^15 + 55*x^14 - 8*x^13 - 55*x^12 + 69*x^11 - 19*x^10 - 37*x^9 + 44*x^8 - 16*x^7 - 6*x^6 + 15*x^5 - 14*x^4 + 3*x^3 + 7*x^2 - 5*x + 1)
gp: K = bnfinit(x^18 - 8*x^17 + 28*x^16 - 54*x^15 + 55*x^14 - 8*x^13 - 55*x^12 + 69*x^11 - 19*x^10 - 37*x^9 + 44*x^8 - 16*x^7 - 6*x^6 + 15*x^5 - 14*x^4 + 3*x^3 + 7*x^2 - 5*x + 1, 1)

## Normalizeddefining polynomial

$$x^{18}$$ $$\mathstrut -\mathstrut 8 x^{17}$$ $$\mathstrut +\mathstrut 28 x^{16}$$ $$\mathstrut -\mathstrut 54 x^{15}$$ $$\mathstrut +\mathstrut 55 x^{14}$$ $$\mathstrut -\mathstrut 8 x^{13}$$ $$\mathstrut -\mathstrut 55 x^{12}$$ $$\mathstrut +\mathstrut 69 x^{11}$$ $$\mathstrut -\mathstrut 19 x^{10}$$ $$\mathstrut -\mathstrut 37 x^{9}$$ $$\mathstrut +\mathstrut 44 x^{8}$$ $$\mathstrut -\mathstrut 16 x^{7}$$ $$\mathstrut -\mathstrut 6 x^{6}$$ $$\mathstrut +\mathstrut 15 x^{5}$$ $$\mathstrut -\mathstrut 14 x^{4}$$ $$\mathstrut +\mathstrut 3 x^{3}$$ $$\mathstrut +\mathstrut 7 x^{2}$$ $$\mathstrut -\mathstrut 5 x$$ $$\mathstrut +\mathstrut 1$$

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: $$-11126075312143749771=-\,3^{4}\cdot 11^{5}\cdot 31^{8}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $11.43$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $3, 11, 31$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{615673} a^{17} - \frac{31358}{615673} a^{16} - \frac{156453}{615673} a^{15} - \frac{265295}{615673} a^{14} - \frac{128252}{615673} a^{13} - \frac{260171}{615673} a^{12} - \frac{75109}{615673} a^{11} - \frac{282006}{615673} a^{10} - \frac{176199}{615673} a^{9} + \frac{20457}{615673} a^{8} + \frac{204360}{615673} a^{7} + \frac{7222}{615673} a^{6} + \frac{157958}{615673} a^{5} - \frac{125346}{615673} a^{4} - \frac{243673}{615673} a^{3} - \frac{122031}{615673} a^{2} - \frac{120165}{615673} a - \frac{130342}{615673}$

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: $$\frac{270389}{615673} a^{17} - \frac{2272398}{615673} a^{16} + \frac{8341035}{615673} a^{15} - \frac{16796023}{615673} a^{14} + \frac{17706214}{615673} a^{13} - \frac{2426558}{615673} a^{12} - \frac{19143686}{615673} a^{11} + \frac{23791963}{615673} a^{10} - \frac{5188709}{615673} a^{9} - \frac{15250284}{615673} a^{8} + \frac{16667461}{615673} a^{7} - \frac{4475109}{615673} a^{6} - \frac{4055732}{615673} a^{5} + \frac{5544440}{615673} a^{4} - \frac{4562413}{615673} a^{3} + \frac{538703}{615673} a^{2} + \frac{3311082}{615673} a - \frac{1920518}{615673}$$,  $$\frac{293260}{615673} a^{17} - \frac{2202171}{615673} a^{16} + \frac{7164602}{615673} a^{15} - \frac{12590842}{615673} a^{14} + \frac{10748491}{615673} a^{13} + \frac{1991757}{615673} a^{12} - \frac{15539917}{615673} a^{11} + \frac{15203663}{615673} a^{10} - \frac{1146542}{615673} a^{9} - \frac{10980006}{615673} a^{8} + \frac{10238875}{615673} a^{7} - \frac{3069765}{615673} a^{6} - \frac{1704786}{615673} a^{5} + \frac{4098216}{615673} a^{4} - \frac{3304254}{615673} a^{3} + \frac{413411}{615673} a^{2} + \frac{2150293}{615673} a - \frac{1268061}{615673}$$,  $$\frac{958357}{615673} a^{17} - \frac{7316406}{615673} a^{16} + \frac{24107581}{615673} a^{15} - \frac{42711018}{615673} a^{14} + \frac{36849117}{615673} a^{13} + \frac{5324896}{615673} a^{12} - \frac{49080958}{615673} a^{11} + \frac{46343816}{615673} a^{10} - \frac{911333}{615673} a^{9} - \frac{34859551}{615673} a^{8} + \frac{28882140}{615673} a^{7} - \frac{5066996}{615673} a^{6} - \frac{7678964}{615673} a^{5} + \frac{12518660}{615673} a^{4} - \frac{8960110}{615673} a^{3} - \frac{945371}{615673} a^{2} + \frac{6206802}{615673} a - \frac{2120143}{615673}$$,  $$\frac{109286}{615673} a^{17} - \frac{770143}{615673} a^{16} + \frac{2195017}{615673} a^{15} - \frac{2834819}{615673} a^{14} + \frac{263446}{615673} a^{13} + \frac{4272291}{615673} a^{12} - \frac{5135122}{615673} a^{11} + \frac{51318}{615673} a^{10} + \frac{5245891}{615673} a^{9} - \frac{4770345}{615673} a^{8} + \frac{148885}{615673} a^{7} + \frac{2433398}{615673} a^{6} - \frac{1488605}{615673} a^{5} + \frac{776967}{615673} a^{4} + \frac{272464}{615673} a^{3} - \frac{1418359}{615673} a^{2} + \frac{568573}{615673} a + \frac{270389}{615673}$$,  $$\frac{470135}{615673} a^{17} - \frac{3897383}{615673} a^{16} + \frac{13966961}{615673} a^{15} - \frac{27286751}{615673} a^{14} + \frac{27886520}{615673} a^{13} - \frac{4047886}{615673} a^{12} - \frac{27150085}{615673} a^{11} + \frac{32610698}{615673} a^{10} - \frac{7134137}{615673} a^{9} - \frac{18346428}{615673} a^{8} + \frac{18871467}{615673} a^{7} - \frac{5047009}{615673} a^{6} - \frac{2739949}{615673} a^{5} + \frac{5756215}{615673} a^{4} - \frac{6471802}{615673} a^{3} + \frac{1059993}{615673} a^{2} + \frac{3460570}{615673} a - \frac{1633826}{615673}$$,  $$a - 1$$,  $$\frac{645300}{615673} a^{17} - \frac{4918293}{615673} a^{16} + \frac{16176484}{615673} a^{15} - \frac{28534405}{615673} a^{14} + \frac{24222999}{615673} a^{13} + \frac{4449454}{615673} a^{12} - \frac{33458463}{615673} a^{11} + \frac{30474498}{615673} a^{10} + \frac{659267}{615673} a^{9} - \frac{23753940}{615673} a^{8} + \frac{18515628}{615673} a^{7} - \frac{2750702}{615673} a^{6} - \frac{4834191}{615673} a^{5} + \frac{8117343}{615673} a^{4} - \frac{6690776}{615673} a^{3} - \frac{180581}{615673} a^{2} + \frac{4618215}{615673} a - \frac{1372724}{615673}$$,  $$\frac{596620}{615673} a^{17} - \frac{4664220}{615673} a^{16} + \frac{15817841}{615673} a^{15} - \frac{28946326}{615673} a^{14} + \frac{25837485}{615673} a^{13} + \frac{3948778}{615673} a^{12} - \frac{37328328}{615673} a^{11} + \frac{37022427}{615673} a^{10} - \frac{1992341}{615673} a^{9} - \frac{27751497}{615673} a^{8} + \frac{23856219}{615673} a^{7} - \frac{3384052}{615673} a^{6} - \frac{7552226}{615673} a^{5} + \frac{9256866}{615673} a^{4} - \frac{6860827}{615673} a^{3} + \frac{275395}{615673} a^{2} + \frac{5356915}{615673} a - \frac{2065775}{615673}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$80.526625464$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A solvable group of order 5184 The 58 conjugacy class representatives for t18n485 are not computed Character table for t18n485 is not computed

## Intermediate fields

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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 $18$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4} 3.6.0.1x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4} 1111.6.5.1x^{6} - 11$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
11.12.0.1$x^{12} - x + 7$$1$$12$$0$$C_{12}$$[\ ]^{12} 31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2} 31.2.0.1x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$V_4$$[\ ]_{2}^{2} 31.8.6.1x^{8} - 7471 x^{4} + 19927296$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$