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Properties

Label 18.0.78582417975...0723.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 7^{12}\cdot 19^{16}$
Root discriminant $86.82$
Ramified primes $3, 7, 19$
Class number $17739$ (GRH)
Class group $[3, 3, 3, 657]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![519841, 277585, 1415022, 1462041, 3326344, 2663843, 2797383, 1297357, 1037476, 362877, 258875, 46050, 23855, 1282, 1551, 32, 47, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 47*x^16 + 32*x^15 + 1551*x^14 + 1282*x^13 + 23855*x^12 + 46050*x^11 + 258875*x^10 + 362877*x^9 + 1037476*x^8 + 1297357*x^7 + 2797383*x^6 + 2663843*x^5 + 3326344*x^4 + 1462041*x^3 + 1415022*x^2 + 277585*x + 519841)
 
gp: K = bnfinit(x^18 - x^17 + 47*x^16 + 32*x^15 + 1551*x^14 + 1282*x^13 + 23855*x^12 + 46050*x^11 + 258875*x^10 + 362877*x^9 + 1037476*x^8 + 1297357*x^7 + 2797383*x^6 + 2663843*x^5 + 3326344*x^4 + 1462041*x^3 + 1415022*x^2 + 277585*x + 519841, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 47 x^{16} + 32 x^{15} + 1551 x^{14} + 1282 x^{13} + 23855 x^{12} + 46050 x^{11} + 258875 x^{10} + 362877 x^{9} + 1037476 x^{8} + 1297357 x^{7} + 2797383 x^{6} + 2663843 x^{5} + 3326344 x^{4} + 1462041 x^{3} + 1415022 x^{2} + 277585 x + 519841 \)

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:  \(-78582417975702211875446018069790723=-\,3^{9}\cdot 7^{12}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.82$
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:  $3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(399=3\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{399}(64,·)$, $\chi_{399}(1,·)$, $\chi_{399}(130,·)$, $\chi_{399}(197,·)$, $\chi_{399}(134,·)$, $\chi_{399}(263,·)$, $\chi_{399}(74,·)$, $\chi_{399}(142,·)$, $\chi_{399}(275,·)$, $\chi_{399}(340,·)$, $\chi_{399}(214,·)$, $\chi_{399}(23,·)$, $\chi_{399}(347,·)$, $\chi_{399}(289,·)$, $\chi_{399}(106,·)$, $\chi_{399}(44,·)$, $\chi_{399}(239,·)$, $\chi_{399}(310,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{1}{7} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{9} + \frac{3}{7} a^{6} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{13} + \frac{3}{7} a^{10} + \frac{3}{7} a^{7} + \frac{1}{7} a^{4}$, $\frac{1}{7} a^{14} + \frac{3}{7} a^{10} - \frac{3}{7} a^{9} - \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{6} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4} - \frac{3}{7} a^{3}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{9} - \frac{1}{7} a^{6} - \frac{3}{7} a^{3}$, $\frac{1}{38554218270323} a^{16} + \frac{83450907868}{1243684460333} a^{15} + \frac{1583196637635}{38554218270323} a^{14} - \frac{2073724444285}{38554218270323} a^{13} + \frac{2416570250834}{38554218270323} a^{12} - \frac{577132090}{23696507849} a^{11} - \frac{12123647067501}{38554218270323} a^{10} - \frac{15836430048189}{38554218270323} a^{9} - \frac{7325091444092}{38554218270323} a^{8} + \frac{284444581209}{1243684460333} a^{7} - \frac{13638646054946}{38554218270323} a^{6} - \frac{3875418346453}{38554218270323} a^{5} + \frac{2223315104162}{5507745467189} a^{4} + \frac{236090779696}{786820781027} a^{3} - \frac{1824157539315}{5507745467189} a^{2} + \frac{198265099671}{786820781027} a - \frac{1733435380}{7639036709}$, $\frac{1}{3319703405243515151312619446377887373728317} a^{17} - \frac{22500218981207812409091778778}{3319703405243515151312619446377887373728317} a^{16} + \frac{153241115673469028788289676021479809905923}{3319703405243515151312619446377887373728317} a^{15} + \frac{26628653628257015007970282876882380223349}{474243343606216450187517063768269624818331} a^{14} + \frac{131066551016598105644635707291803408419527}{3319703405243515151312619446377887373728317} a^{13} - \frac{23194386841362526403148044121809381817}{656976727734715050724840579136728156289} a^{12} - \frac{7536697882311511131265042911194826593927}{3319703405243515151312619446377887373728317} a^{11} - \frac{638008361874212960354284692338973869964391}{3319703405243515151312619446377887373728317} a^{10} + \frac{713662185817297970234357983917659936237577}{3319703405243515151312619446377887373728317} a^{9} + \frac{1349148252179869035390471018176775193628264}{3319703405243515151312619446377887373728317} a^{8} - \frac{1440612275092295948494411455690113969625}{10674287476667251290394274747195779336747} a^{7} + \frac{222458154912542561419512320309073236534110}{3319703405243515151312619446377887373728317} a^{6} + \frac{895165533924802469523199671571927193344246}{3319703405243515151312619446377887373728317} a^{5} + \frac{173937342158175125321747014653282128605177}{474243343606216450187517063768269624818331} a^{4} - \frac{67273508468415303114504061102360650893110}{474243343606216450187517063768269624818331} a^{3} + \frac{61888138820347973871928034785406152046396}{474243343606216450187517063768269624818331} a^{2} - \frac{8419837839141315446116561650481944875523}{67749049086602350026788151966895660688333} a + \frac{48949429523923863729060739255808034647}{657757758122352912881438368610637482411}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{657}$, which has order $17739$ (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:  \( \frac{7589917245105410925492847502}{13471699310111139523611785648995303} a^{17} - \frac{19169991991262943032767555806}{13471699310111139523611785648995303} a^{16} + \frac{363867344065414079614941516718}{13471699310111139523611785648995303} a^{15} - \frac{295834634669206169794989624205}{13471699310111139523611785648995303} a^{14} + \frac{11199545276055580370786915128474}{13471699310111139523611785648995303} a^{13} - \frac{1647854981410021283370180833}{2666079420168442415122063259251} a^{12} + \frac{159722910259045717187060475747382}{13471699310111139523611785648995303} a^{11} + \frac{69612751514510115194437663487351}{13471699310111139523611785648995303} a^{10} + \frac{1339007348736051779138456240150072}{13471699310111139523611785648995303} a^{9} - \frac{410460520496387495013598948461692}{13471699310111139523611785648995303} a^{8} + \frac{2750261658641188499751358676074096}{13471699310111139523611785648995303} a^{7} - \frac{3137838667506079253179069897918916}{13471699310111139523611785648995303} a^{6} + \frac{3899458001083404079832563149227049}{13471699310111139523611785648995303} a^{5} - \frac{2057314012321689229920289797222804}{1924528472873019931944540806999329} a^{4} - \frac{1538937134767202099169041637750363}{1924528472873019931944540806999329} a^{3} - \frac{4621651767711229762028589105103453}{1924528472873019931944540806999329} a^{2} - \frac{75642335194072262023916839994873}{274932638981859990277791543857047} a - \frac{537770729348345023326954686571}{2669248922153980488133898484049} \) (order $6$)
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:  \( 4369063.34801 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.361.1, 6.0.3518667.1, 9.9.1998099208210609.2

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 $18$ R $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ R $18$ $18$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ $18$ $18$

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
3Data not computed
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
$19$19.9.8.6$x^{9} + 1216$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.6$x^{9} + 1216$$9$$1$$8$$C_9$$[\ ]_{9}$