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Properties

Label 18.0.42004352444...8672.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 7^{12}\cdot 19^{15}$
Root discriminant $442.32$
Ramified primes $2, 3, 7, 19$
Class number $1013166336$ (GRH)
Class group $[2, 2, 2, 2, 28, 2261532]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2551664661337, 3640164079860, 2224622720133, 615461531370, 7539888843, -42861762090, -9047417835, 590724540, 452260827, 39238850, -5355135, -1285200, -53472, 10080, 3024, -210, 45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 45*x^16 - 210*x^15 + 3024*x^14 + 10080*x^13 - 53472*x^12 - 1285200*x^11 - 5355135*x^10 + 39238850*x^9 + 452260827*x^8 + 590724540*x^7 - 9047417835*x^6 - 42861762090*x^5 + 7539888843*x^4 + 615461531370*x^3 + 2224622720133*x^2 + 3640164079860*x + 2551664661337)
 
gp: K = bnfinit(x^18 + 45*x^16 - 210*x^15 + 3024*x^14 + 10080*x^13 - 53472*x^12 - 1285200*x^11 - 5355135*x^10 + 39238850*x^9 + 452260827*x^8 + 590724540*x^7 - 9047417835*x^6 - 42861762090*x^5 + 7539888843*x^4 + 615461531370*x^3 + 2224622720133*x^2 + 3640164079860*x + 2551664661337, 1)
 

Normalized defining polynomial

\( x^{18} + 45 x^{16} - 210 x^{15} + 3024 x^{14} + 10080 x^{13} - 53472 x^{12} - 1285200 x^{11} - 5355135 x^{10} + 39238850 x^{9} + 452260827 x^{8} + 590724540 x^{7} - 9047417835 x^{6} - 42861762090 x^{5} + 7539888843 x^{4} + 615461531370 x^{3} + 2224622720133 x^{2} + 3640164079860 x + 2551664661337 \)

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:  \(-420043524447982694779997090972698756380285468672=-\,2^{18}\cdot 3^{27}\cdot 7^{12}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $442.32$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4788=2^{2}\cdot 3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4788}(1,·)$, $\chi_{4788}(3371,·)$, $\chi_{4788}(961,·)$, $\chi_{4788}(2843,·)$, $\chi_{4788}(457,·)$, $\chi_{4788}(3469,·)$, $\chi_{4788}(3599,·)$, $\chi_{4788}(2963,·)$, $\chi_{4788}(2965,·)$, $\chi_{4788}(2459,·)$, $\chi_{4788}(3875,·)$, $\chi_{4788}(1703,·)$, $\chi_{4788}(4225,·)$, $\chi_{4788}(4103,·)$, $\chi_{4788}(1261,·)$, $\chi_{4788}(1717,·)$, $\chi_{4788}(2615,·)$, $\chi_{4788}(505,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{9} a^{5} - \frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{4}{9} a^{2} - \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{27} a^{6} + \frac{1}{9} a^{4} + \frac{2}{27} a^{3} + \frac{1}{3} a^{2} + \frac{1}{9} a + \frac{1}{27}$, $\frac{1}{27} a^{7} - \frac{4}{27} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{4}{27} a + \frac{2}{9}$, $\frac{1}{54} a^{8} + \frac{1}{27} a^{5} - \frac{1}{6} a^{4} + \frac{1}{9} a^{3} + \frac{1}{54} a^{2} - \frac{1}{3} a - \frac{7}{18}$, $\frac{1}{162} a^{9} - \frac{1}{18} a^{5} + \frac{5}{54} a^{3} + \frac{1}{9} a^{2} + \frac{1}{6} a + \frac{8}{81}$, $\frac{1}{1782} a^{10} + \frac{2}{891} a^{9} - \frac{1}{297} a^{8} - \frac{4}{297} a^{7} + \frac{1}{594} a^{6} + \frac{13}{297} a^{5} - \frac{83}{594} a^{4} - \frac{1}{297} a^{3} + \frac{91}{594} a^{2} - \frac{4}{891} a + \frac{28}{81}$, $\frac{1}{5346} a^{11} - \frac{1}{5346} a^{10} + \frac{7}{5346} a^{9} + \frac{13}{1782} a^{8} - \frac{25}{1782} a^{7} - \frac{23}{1782} a^{6} - \frac{46}{891} a^{5} - \frac{74}{891} a^{4} + \frac{122}{891} a^{3} + \frac{1112}{2673} a^{2} + \frac{1547}{5346} a - \frac{145}{486}$, $\frac{1}{112266} a^{12} - \frac{1}{37422} a^{10} + \frac{25}{16038} a^{9} + \frac{4}{2079} a^{8} + \frac{10}{891} a^{7} - \frac{23}{2673} a^{6} - \frac{29}{594} a^{5} + \frac{479}{12474} a^{4} + \frac{1123}{16038} a^{3} - \frac{632}{2079} a^{2} - \frac{1201}{5346} a - \frac{3379}{10206}$, $\frac{1}{112266} a^{13} - \frac{1}{37422} a^{11} - \frac{1}{8019} a^{10} + \frac{17}{12474} a^{9} + \frac{5}{1782} a^{8} - \frac{14}{2673} a^{7} - \frac{5}{297} a^{6} + \frac{229}{6237} a^{5} + \frac{421}{16038} a^{4} + \frac{395}{4158} a^{3} - \frac{193}{5346} a^{2} - \frac{23026}{56133} a + \frac{25}{162}$, $\frac{1}{785862} a^{14} - \frac{1}{785862} a^{12} + \frac{1}{112266} a^{11} - \frac{1}{6237} a^{10} + \frac{85}{56133} a^{9} + \frac{2111}{261954} a^{8} + \frac{7}{594} a^{7} + \frac{1381}{261954} a^{6} - \frac{1811}{112266} a^{5} - \frac{2543}{29106} a^{4} - \frac{2455}{112266} a^{3} - \frac{341}{35721} a^{2} - \frac{212}{567} a + \frac{12499}{71442}$, $\frac{1}{2357586} a^{15} + \frac{1}{392931} a^{13} - \frac{1}{336798} a^{12} + \frac{13}{56133} a^{10} - \frac{316}{1178793} a^{9} + \frac{2}{2079} a^{8} + \frac{2231}{130977} a^{7} - \frac{3407}{336798} a^{6} + \frac{394}{14553} a^{5} + \frac{1438}{56133} a^{4} + \frac{40951}{1178793} a^{3} + \frac{151}{4158} a^{2} - \frac{36733}{785862} a - \frac{3217}{30618}$, $\frac{1}{158494599379061421623532} a^{16} + \frac{4274416331506813}{79247299689530710811766} a^{15} - \frac{8043975378561074}{13207883281588451801961} a^{14} + \frac{4142287766563357}{79247299689530710811766} a^{13} + \frac{186197852219625689}{79247299689530710811766} a^{12} + \frac{9854036595886756}{269548638399764322489} a^{11} + \frac{9133456333504819438}{39623649844765355405883} a^{10} - \frac{77013256228738308145}{39623649844765355405883} a^{9} + \frac{3005603697885535547}{359398184533019096652} a^{8} - \frac{1210312991807169334429}{79247299689530710811766} a^{7} - \frac{11920526279383012946}{3602149985887759582353} a^{6} + \frac{588281660486549526661}{13207883281588451801961} a^{5} + \frac{11448581876764232449481}{158494599379061421623532} a^{4} - \frac{11660042389692520855993}{79247299689530710811766} a^{3} + \frac{32394902942809266835}{13207883281588451801961} a^{2} + \frac{778424512286138687771}{3602149985887759582353} a - \frac{450389828508663808279}{1309872722141003484492}$, $\frac{1}{17192718888557394043773277361622396} a^{17} - \frac{11075205499}{17192718888557394043773277361622396} a^{16} + \frac{364822543638996878321311393}{4298179722139348510943319340405599} a^{15} - \frac{1587547559920212855743952424}{4298179722139348510943319340405599} a^{14} + \frac{27236746365870727617273129197}{8596359444278697021886638680811198} a^{13} - \frac{12581279923721674654447699361}{8596359444278697021886638680811198} a^{12} + \frac{464626783176456965404053069215}{8596359444278697021886638680811198} a^{11} - \frac{1783246366379161403641222000769}{8596359444278697021886638680811198} a^{10} + \frac{29605513830920612734196872047133}{17192718888557394043773277361622396} a^{9} + \frac{16822042053369520338915402190159}{17192718888557394043773277361622396} a^{8} + \frac{9193137812349329368167631909583}{781487222207154274716967152801018} a^{7} - \frac{26043439917459919010468432858305}{8596359444278697021886638680811198} a^{6} - \frac{121553256185460005060306579601529}{2456102698365342006253325337374628} a^{5} - \frac{2561596053467660224045345294931885}{17192718888557394043773277361622396} a^{4} - \frac{10684138617160283091641661656749}{614025674591335501563331334343657} a^{3} - \frac{327762645973860144683401732987079}{781487222207154274716967152801018} a^{2} - \frac{454395646986620479596959831432507}{1562974444414308549433934305602036} a + \frac{27597258582852752477928515726017}{142088585855846231766721300509276}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{28}\times C_{2261532}$, which has order $1013166336$ (assuming GRH)

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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 110172188.8644179 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-57}) \), 3.3.361.1, 3.3.3969.2, 3.3.1432809.2, 3.3.1432809.3, 6.0.4278699072.1, 6.0.20745515423808.9, 6.0.7489131067994688.4, 6.0.7489131067994688.2, 9.9.2941473244627851129.9

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$7$7.6.4.2$x^{6} - 7 x^{3} + 147$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.2$x^{6} - 7 x^{3} + 147$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.2$x^{6} - 7 x^{3} + 147$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19Data not computed