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

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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)

Normalized defining 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

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

9.1.35050633.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ $18$ ${\href{/LocalNumberField/11.14.0.1}{14} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ 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$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
$83$83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.6.0.1$x^{6} - x + 34$$1$$6$$0$$C_6$$[\ ]^{6}$
83.6.0.1$x^{6} - x + 34$$1$$6$$0$$C_6$$[\ ]^{6}$
$107$107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
107.4.0.1$x^{4} - x + 2$$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