↑

Properties

Label 18.0.23267717097...0976.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 7^{9}\cdot 19^{16}$
Root discriminant $376.64$
Ramified primes $2, 3, 7, 19$
Class number $909840384$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 12, 296172]$ (GRH)
Galois group $C_{18}$ (as 18T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29491760233189, 5208110998344, 5532391423872, 584216479326, 434832004497, 26380180944, 21570475153, 779478990, 775572138, 13599706, 20079240, 107388, 413371, 2052, 6471, -76, 75, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 75*x^16 - 76*x^15 + 6471*x^14 + 2052*x^13 + 413371*x^12 + 107388*x^11 + 20079240*x^10 + 13599706*x^9 + 775572138*x^8 + 779478990*x^7 + 21570475153*x^6 + 26380180944*x^5 + 434832004497*x^4 + 584216479326*x^3 + 5532391423872*x^2 + 5208110998344*x + 29491760233189)
 
gp: K = bnfinit(x^18 + 75*x^16 - 76*x^15 + 6471*x^14 + 2052*x^13 + 413371*x^12 + 107388*x^11 + 20079240*x^10 + 13599706*x^9 + 775572138*x^8 + 779478990*x^7 + 21570475153*x^6 + 26380180944*x^5 + 434832004497*x^4 + 584216479326*x^3 + 5532391423872*x^2 + 5208110998344*x + 29491760233189, 1)
 

Normalized defining polynomial

\( x^{18} + 75 x^{16} - 76 x^{15} + 6471 x^{14} + 2052 x^{13} + 413371 x^{12} + 107388 x^{11} + 20079240 x^{10} + 13599706 x^{9} + 775572138 x^{8} + 779478990 x^{7} + 21570475153 x^{6} + 26380180944 x^{5} + 434832004497 x^{4} + 584216479326 x^{3} + 5532391423872 x^{2} + 5208110998344 x + 29491760233189 \)

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:  \(-23267717097701665308512958392073692044388990976=-\,2^{18}\cdot 3^{27}\cdot 7^{9}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $376.64$
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$$\chi_{4788}(1,·)$, $\chi_{4788}(3275,·)$, $\chi_{4788}(2437,·)$, $\chi_{4788}(587,·)$, $\chi_{4788}(4621,·)$, $\chi_{4788}(1261,·)$, $\chi_{4788}(85,·)$, $\chi_{4788}(2519,·)$, $\chi_{4788}(4367,·)$, $\chi_{4788}(2015,·)$, $\chi_{4788}(1849,·)$, $\chi_{4788}(2855,·)$, $\chi_{4788}(169,·)$, $\chi_{4788}(3949,·)$, $\chi_{4788}(3695,·)$, $\chi_{4788}(3443,·)$, $\chi_{4788}(505,·)$, $\chi_{4788}(671,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{21} a^{10} + \frac{2}{21} a^{9} + \frac{1}{7} a^{8} + \frac{4}{21} a^{7} + \frac{5}{21} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a^{2} + \frac{2}{21} a + \frac{1}{21}$, $\frac{1}{21} a^{11} - \frac{1}{21} a^{9} - \frac{2}{21} a^{8} - \frac{1}{7} a^{7} - \frac{4}{21} a^{6} - \frac{2}{7} a^{4} + \frac{3}{7} a^{3} - \frac{4}{21} a^{2} - \frac{1}{7} a - \frac{2}{21}$, $\frac{1}{147} a^{12} - \frac{2}{147} a^{11} - \frac{1}{49} a^{10} + \frac{17}{147} a^{9} - \frac{5}{147} a^{8} - \frac{23}{49} a^{7} + \frac{61}{147} a^{6} + \frac{22}{49} a^{5} + \frac{13}{49} a^{4} - \frac{16}{147} a^{3} + \frac{41}{147} a^{2} + \frac{1}{7} a - \frac{19}{147}$, $\frac{1}{5439} a^{13} + \frac{13}{5439} a^{12} - \frac{110}{5439} a^{11} + \frac{2}{259} a^{10} - \frac{905}{5439} a^{9} - \frac{1691}{5439} a^{8} - \frac{169}{5439} a^{7} - \frac{107}{1813} a^{6} - \frac{57}{259} a^{5} + \frac{401}{5439} a^{4} - \frac{1984}{5439} a^{3} + \frac{2036}{5439} a^{2} - \frac{2567}{5439} a - \frac{16}{49}$, $\frac{1}{38073} a^{14} + \frac{1}{38073} a^{13} - \frac{1}{5439} a^{12} - \frac{64}{12691} a^{11} + \frac{404}{38073} a^{10} + \frac{5284}{38073} a^{9} - \frac{9403}{38073} a^{8} + \frac{13621}{38073} a^{7} - \frac{6151}{38073} a^{6} + \frac{3887}{38073} a^{5} - \frac{2911}{38073} a^{4} - \frac{17}{777} a^{3} + \frac{139}{5439} a^{2} + \frac{10121}{38073} a + \frac{226}{1029}$, $\frac{1}{114219} a^{15} - \frac{1}{114219} a^{14} - \frac{2}{114219} a^{13} - \frac{346}{114219} a^{12} + \frac{261}{12691} a^{11} + \frac{1921}{114219} a^{10} - \frac{2056}{16317} a^{9} - \frac{30692}{114219} a^{8} - \frac{4072}{12691} a^{7} - \frac{13310}{38073} a^{6} - \frac{19841}{114219} a^{5} + \frac{19451}{114219} a^{4} - \frac{367}{2331} a^{3} + \frac{31751}{114219} a^{2} - \frac{3302}{38073} a + \frac{521}{3087}$, $\frac{1}{480519333} a^{16} + \frac{1916}{480519333} a^{15} + \frac{6175}{480519333} a^{14} + \frac{25838}{480519333} a^{13} - \frac{8165}{3268839} a^{12} - \frac{1555217}{68645619} a^{11} - \frac{1490011}{68645619} a^{10} + \frac{16054774}{480519333} a^{9} + \frac{20157671}{53391037} a^{8} + \frac{39382919}{160173111} a^{7} + \frac{135994897}{480519333} a^{6} + \frac{3831938}{480519333} a^{5} - \frac{159644866}{480519333} a^{4} + \frac{50109674}{480519333} a^{3} - \frac{2539766}{22881873} a^{2} + \frac{112289882}{480519333} a + \frac{1696378}{4329003}$, $\frac{1}{84099196868999726900542715725934931012124918215458406661330728211539} a^{17} - \frac{19176551359533999320628286449678738824987509263067930839767}{84099196868999726900542715725934931012124918215458406661330728211539} a^{16} + \frac{235253947992459374406835117170450768954273715576125639815502710}{84099196868999726900542715725934931012124918215458406661330728211539} a^{15} + \frac{474143943302389880088764270562638735812949999703602814632980501}{84099196868999726900542715725934931012124918215458406661330728211539} a^{14} + \frac{7409637965996755840266876113227483539121581867078750211260304207}{84099196868999726900542715725934931012124918215458406661330728211539} a^{13} + \frac{6225619168210543012770981785516587161983496830408105028608355625}{12014170981285675271506102246562133001732131173636915237332961173077} a^{12} - \frac{40519636791316295860826592329566505067565987095347793531961036163}{12014170981285675271506102246562133001732131173636915237332961173077} a^{11} - \frac{92387413324614944210073723523607653761556558828215849865516071742}{84099196868999726900542715725934931012124918215458406661330728211539} a^{10} + \frac{13030508177127397784390087790210338563929737288922477879993183375656}{84099196868999726900542715725934931012124918215458406661330728211539} a^{9} - \frac{3354232339185884153539690075194097588341009315637820832401713229245}{28033065622999908966847571908644977004041639405152802220443576070513} a^{8} - \frac{6828779413576617917933776438019662156131116704863619063259327015813}{84099196868999726900542715725934931012124918215458406661330728211539} a^{7} - \frac{40468938312832849271759727388604277584626879765900378973533495467554}{84099196868999726900542715725934931012124918215458406661330728211539} a^{6} + \frac{2310932117360816620113980372793220601889114710592382771000611687596}{9344355207666636322282523969548325668013879801717600740147858690171} a^{5} - \frac{858532842044415997833284680643708103718826690991763399549843408412}{28033065622999908966847571908644977004041639405152802220443576070513} a^{4} + \frac{29188313151299954993357567519267543143025862342042903127842467468842}{84099196868999726900542715725934931012124918215458406661330728211539} a^{3} - \frac{36490711107694215648575501741755314686639425156629445076855035931890}{84099196868999726900542715725934931012124918215458406661330728211539} a^{2} - \frac{32430618893456979703255241865775110862391167387309577777968788661229}{84099196868999726900542715725934931012124918215458406661330728211539} a - \frac{109222089894744823043511543445878162724174290521387649200040683056}{757650422243240782887772213747161540649774037977102762714691245149}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{12}\times C_{296172}$, which has order $909840384$ (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:  \( 22027035.20428972 \) (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

\(\Q(\sqrt{-21}) \), 3.3.361.1, 6.0.77241777984.6, 9.9.9025761726072081.2

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 R R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ $18$ $18$ $18$

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
2Data not computed
3Data not computed
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
$19$19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$