# Properties

 Label 18.2.10910724785...9009.1 Degree $18$ Signature $[2, 8]$ Discriminant $43^{2}\cdot 83\cdot 107\cdot 311^{2}\cdot 2621^{2}$ Root discriminant $11.42$ Ramified primes $43, 83, 107, 311, 2621$ Class number $1$ Class group Trivial Galois Group 18T968

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 11, -10, 2, 7, -2, -22, 20, 1, -10, 2, 7, 1, -7, 4, 0, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 4*x^15 - 7*x^14 + x^13 + 7*x^12 + 2*x^11 - 10*x^10 + x^9 + 20*x^8 - 22*x^7 - 2*x^6 + 7*x^5 + 2*x^4 - 10*x^3 + 11*x^2 - 5*x + 1)
gp: K = bnfinit(x^18 - 2*x^17 + 4*x^15 - 7*x^14 + x^13 + 7*x^12 + 2*x^11 - 10*x^10 + x^9 + 20*x^8 - 22*x^7 - 2*x^6 + 7*x^5 + 2*x^4 - 10*x^3 + 11*x^2 - 5*x + 1, 1)

## Normalizeddefining polynomial

$$x^{18}$$ $$\mathstrut -\mathstrut 2 x^{17}$$ $$\mathstrut +\mathstrut 4 x^{15}$$ $$\mathstrut -\mathstrut 7 x^{14}$$ $$\mathstrut +\mathstrut x^{13}$$ $$\mathstrut +\mathstrut 7 x^{12}$$ $$\mathstrut +\mathstrut 2 x^{11}$$ $$\mathstrut -\mathstrut 10 x^{10}$$ $$\mathstrut +\mathstrut x^{9}$$ $$\mathstrut +\mathstrut 20 x^{8}$$ $$\mathstrut -\mathstrut 22 x^{7}$$ $$\mathstrut -\mathstrut 2 x^{6}$$ $$\mathstrut +\mathstrut 7 x^{5}$$ $$\mathstrut +\mathstrut 2 x^{4}$$ $$\mathstrut -\mathstrut 10 x^{3}$$ $$\mathstrut +\mathstrut 11 x^{2}$$ $$\mathstrut -\mathstrut 5 x$$ $$\mathstrut +\mathstrut 1$$

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

## Invariants

 Degree: $18$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[2, 8]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $$10910724785335819009=43^{2}\cdot 83\cdot 107\cdot 311^{2}\cdot 2621^{2}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $11.42$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $43, 83, 107, 311, 2621$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2536123627} a^{17} + \frac{333588062}{2536123627} a^{16} - \frac{456392292}{2536123627} a^{15} + \frac{355601592}{2536123627} a^{14} + \frac{455066414}{2536123627} a^{13} - \frac{6296606}{2536123627} a^{12} + \frac{777290417}{2536123627} a^{11} - \frac{998120764}{2536123627} a^{10} + \frac{139136720}{2536123627} a^{9} - \frac{134210511}{2536123627} a^{8} + \frac{239163599}{2536123627} a^{7} - \frac{708830024}{2536123627} a^{6} + \frac{877414092}{2536123627} a^{5} + \frac{1027607358}{2536123627} a^{4} - \frac{109031228}{2536123627} a^{3} - \frac{1066153715}{2536123627} a^{2} - \frac{1205708747}{2536123627} a + \frac{168602202}{2536123627}$

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: $9$ 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: $$\frac{1083118396}{2536123627} a^{17} - \frac{1183964211}{2536123627} a^{16} - \frac{1517579809}{2536123627} a^{15} + \frac{3430132757}{2536123627} a^{14} - \frac{3867406756}{2536123627} a^{13} - \frac{3661889466}{2536123627} a^{12} + \frac{5794509444}{2536123627} a^{11} + \frac{8604894738}{2536123627} a^{10} - \frac{4780863761}{2536123627} a^{9} - \frac{7116079219}{2536123627} a^{8} + \frac{16443708488}{2536123627} a^{7} - \frac{5626051194}{2536123627} a^{6} - \frac{12845253676}{2536123627} a^{5} - \frac{1909889562}{2536123627} a^{4} + \frac{3087827905}{2536123627} a^{3} - \frac{3318867465}{2536123627} a^{2} + \frac{6133863175}{2536123627} a + \frac{809186095}{2536123627}$$,  $$\frac{147413056}{2536123627} a^{17} - \frac{1109225995}{2536123627} a^{16} + \frac{1097422941}{2536123627} a^{15} + \frac{1387359224}{2536123627} a^{14} - \frac{3921847727}{2536123627} a^{13} + \frac{4097852302}{2536123627} a^{12} + \frac{3012121435}{2536123627} a^{11} - \frac{4760663814}{2536123627} a^{10} - \frac{6112521924}{2536123627} a^{9} + \frac{5068598702}{2536123627} a^{8} + \frac{5712258616}{2536123627} a^{7} - \frac{18652715257}{2536123627} a^{6} + \frac{7991174168}{2536123627} a^{5} + \frac{10830941746}{2536123627} a^{4} - \frac{1363643689}{2536123627} a^{3} - \frac{3796841817}{2536123627} a^{2} + \frac{8583636394}{2536123627} a - \frac{2939115562}{2536123627}$$,  $$\frac{1762172871}{2536123627} a^{17} - \frac{2401810859}{2536123627} a^{16} - \frac{1923409908}{2536123627} a^{15} + \frac{6428923864}{2536123627} a^{14} - \frac{7746453945}{2536123627} a^{13} - \frac{5092538698}{2536123627} a^{12} + \frac{11251286610}{2536123627} a^{11} + \frac{12071635564}{2536123627} a^{10} - \frac{14101060916}{2536123627} a^{9} - \frac{8491019686}{2536123627} a^{8} + \frac{34219460779}{2536123627} a^{7} - \frac{15453058002}{2536123627} a^{6} - \frac{22365707582}{2536123627} a^{5} + \frac{2933824776}{2536123627} a^{4} + \frac{12448100469}{2536123627} a^{3} - \frac{16180969586}{2536123627} a^{2} + \frac{7019377771}{2536123627} a + \frac{308231335}{2536123627}$$,  $$\frac{1594588243}{2536123627} a^{17} - \frac{1528984308}{2536123627} a^{16} - \frac{2097508841}{2536123627} a^{15} + \frac{4649523839}{2536123627} a^{14} - \frac{5664160352}{2536123627} a^{13} - \frac{5820346528}{2536123627} a^{12} + \frac{7027392323}{2536123627} a^{11} + \frac{12491308747}{2536123627} a^{10} - \frac{5533672401}{2536123627} a^{9} - \frac{7389691532}{2536123627} a^{8} + \frac{25877800880}{2536123627} a^{7} - \frac{7195103546}{2536123627} a^{6} - \frac{18612347609}{2536123627} a^{5} - \frac{5044162678}{2536123627} a^{4} + \frac{3451518412}{2536123627} a^{3} - \frac{12083702652}{2536123627} a^{2} + \frac{6753113846}{2536123627} a + \frac{1471512250}{2536123627}$$,  $$a$$,  $$\frac{108329068}{2536123627} a^{17} - \frac{39580205}{2536123627} a^{16} - \frac{823865411}{2536123627} a^{15} + \frac{782859870}{2536123627} a^{14} + \frac{521335225}{2536123627} a^{13} - \frac{2329443423}{2536123627} a^{12} + \frac{2331370078}{2536123627} a^{11} + \frac{3305483114}{2536123627} a^{10} - \frac{1787133582}{2536123627} a^{9} - \frac{5325087427}{2536123627} a^{8} + \frac{2878524887}{2536123627} a^{7} + \frac{3937720658}{2536123627} a^{6} - \frac{11806568359}{2536123627} a^{5} - \frac{839422476}{2536123627} a^{4} + \frac{6468641539}{2536123627} a^{3} + \frac{1956941922}{2536123627} a^{2} - \frac{1220297730}{2536123627} a + \frac{3755278621}{2536123627}$$,  $$\frac{1634111690}{2536123627} a^{17} - \frac{2412626748}{2536123627} a^{16} - \frac{1611676476}{2536123627} a^{15} + \frac{6708393435}{2536123627} a^{14} - \frac{8032473109}{2536123627} a^{13} - \frac{4654597154}{2536123627} a^{12} + \frac{12094558334}{2536123627} a^{11} + \frac{9070366555}{2536123627} a^{10} - \frac{15988120487}{2536123627} a^{9} - \frac{7066383948}{2536123627} a^{8} + \frac{36880608297}{2536123627} a^{7} - \frac{16917828440}{2536123627} a^{6} - \frac{23820386950}{2536123627} a^{5} + \frac{10682869992}{2536123627} a^{4} + \frac{13601357832}{2536123627} a^{3} - \frac{17595703715}{2536123627} a^{2} + \frac{3944952937}{2536123627} a + \frac{752691234}{2536123627}$$,  $$\frac{797401991}{2536123627} a^{17} - \frac{1845181748}{2536123627} a^{16} + \frac{642812823}{2536123627} a^{15} + \frac{3288893602}{2536123627} a^{14} - \frac{6790583042}{2536123627} a^{13} + \frac{2631465109}{2536123627} a^{12} + \frac{5195224123}{2536123627} a^{11} - \frac{826069961}{2536123627} a^{10} - \frac{9278351814}{2536123627} a^{9} + \frac{4154533530}{2536123627} a^{8} + \frac{17751394367}{2536123627} a^{7} - \frac{23435584234}{2536123627} a^{6} + \frac{5264267125}{2536123627} a^{5} + \frac{8758951661}{2536123627} a^{4} + \frac{586968859}{2536123627} a^{3} - \frac{9716083390}{2536123627} a^{2} + \frac{10545144915}{2536123627} a - \frac{6379915334}{2536123627}$$,  $$\frac{532301530}{2536123627} a^{17} + \frac{1034412700}{2536123627} a^{16} - \frac{2413742495}{2536123627} a^{15} - \frac{396739271}{2536123627} a^{14} + \frac{3225412477}{2536123627} a^{13} - \frac{7816883774}{2536123627} a^{12} - \frac{3060855368}{2536123627} a^{11} + \frac{12505559928}{2536123627} a^{10} + \frac{9435167019}{2536123627} a^{9} - \frac{10943751153}{2536123627} a^{8} + \frac{209594279}{2536123627} a^{7} + \frac{24747288166}{2536123627} a^{6} - \frac{14051259987}{2536123627} a^{5} - \frac{20733662495}{2536123627} a^{4} + \frac{518886070}{2536123627} a^{3} + \frac{3199135220}{2536123627} a^{2} - \frac{7771445813}{2536123627} a + \frac{5893201583}{2536123627}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$106.811954559$$ 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 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

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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ $18$ ${\href{/LocalNumberField/11.14.0.1}{14} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$43$$\Q_{43}$$x + 9$$1$$1$$0Trivial[\ ] \Q_{43}$$x + 9$$1$$1$$0Trivial[\ ] 43.2.0.1x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2} 43.4.0.1x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$V_4$$[\ ]_{2}^{2} 43.4.0.1x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
$83$83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2} 83.2.0.1x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 83.6.0.1x^{6} - x + 34$$1$$6$$0$$C_6$$[\ ]^{6}$
83.6.0.1$x^{6} - x + 34$$1$$6$$0$$C_6$$[\ ]^{6} 107107.2.1.2x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4} 107.4.0.1x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
107.8.0.1$x^{8} - x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
311Data not computed
2621Data not computed