↑

Properties

Label 18.0.16373578698...5872.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{24}\cdot 7^{9}\cdot 19^{17}$
Root discriminant $369.36$
Ramified primes $2, 3, 7, 19$
Class number $147122112$ (GRH)
Class group $[2, 2, 4, 9195132]$ (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![18869246125489, -10915804281390, 6005339466180, -1052376262422, 270199015086, 3554059440, 21949850871, -2931191484, 1560486036, 100728424, 36134124, 1799718, 3094283, 59052, 44460, -76, 285, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 285*x^16 - 76*x^15 + 44460*x^14 + 59052*x^13 + 3094283*x^12 + 1799718*x^11 + 36134124*x^10 + 100728424*x^9 + 1560486036*x^8 - 2931191484*x^7 + 21949850871*x^6 + 3554059440*x^5 + 270199015086*x^4 - 1052376262422*x^3 + 6005339466180*x^2 - 10915804281390*x + 18869246125489)
 
gp: K = bnfinit(x^18 + 285*x^16 - 76*x^15 + 44460*x^14 + 59052*x^13 + 3094283*x^12 + 1799718*x^11 + 36134124*x^10 + 100728424*x^9 + 1560486036*x^8 - 2931191484*x^7 + 21949850871*x^6 + 3554059440*x^5 + 270199015086*x^4 - 1052376262422*x^3 + 6005339466180*x^2 - 10915804281390*x + 18869246125489, 1)
 

Normalized defining polynomial

\( x^{18} + 285 x^{16} - 76 x^{15} + 44460 x^{14} + 59052 x^{13} + 3094283 x^{12} + 1799718 x^{11} + 36134124 x^{10} + 100728424 x^{9} + 1560486036 x^{8} - 2931191484 x^{7} + 21949850871 x^{6} + 3554059440 x^{5} + 270199015086 x^{4} - 1052376262422 x^{3} + 6005339466180 x^{2} - 10915804281390 x + 18869246125489 \)

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:  \(-16373578698382653365249859609237042549755215872=-\,2^{18}\cdot 3^{24}\cdot 7^{9}\cdot 19^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $369.36$
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$$\chi_{4788}(1,·)$, $\chi_{4788}(1903,·)$, $\chi_{4788}(841,·)$, $\chi_{4788}(1231,·)$, $\chi_{4788}(1681,·)$, $\chi_{4788}(979,·)$, $\chi_{4788}(1429,·)$, $\chi_{4788}(3415,·)$, $\chi_{4788}(4591,·)$, $\chi_{4788}(3361,·)$, $\chi_{4788}(4003,·)$, $\chi_{4788}(1063,·)$, $\chi_{4788}(1261,·)$, $\chi_{4788}(559,·)$, $\chi_{4788}(2353,·)$, $\chi_{4788}(3445,·)$, $\chi_{4788}(505,·)$, $\chi_{4788}(895,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19} a^{9}$, $\frac{1}{19} a^{10}$, $\frac{1}{19} a^{11}$, $\frac{1}{19} a^{12}$, $\frac{1}{19} a^{13}$, $\frac{1}{19} a^{14}$, $\frac{1}{19} a^{15}$, $\frac{1}{429077} a^{16} - \frac{1469}{429077} a^{15} + \frac{7633}{429077} a^{14} - \frac{10073}{429077} a^{13} + \frac{3503}{429077} a^{12} - \frac{7592}{429077} a^{11} - \frac{1813}{429077} a^{10} + \frac{4226}{429077} a^{9} - \frac{7281}{22583} a^{8} - \frac{6129}{22583} a^{7} + \frac{2070}{22583} a^{6} - \frac{3753}{22583} a^{5} - \frac{2437}{22583} a^{4} + \frac{8667}{22583} a^{3} - \frac{2319}{22583} a^{2} + \frac{2778}{22583} a + \frac{7838}{22583}$, $\frac{1}{38302460064194734332856012656541815136528577022762039872092298880794076157212570123194303598557681} a^{17} + \frac{114361550552225986908762115843697881087093704517548701754196258886224491974086286456166643}{101061899905527003516770481943382097985563527764543640823462530028480412024307572884417687595139} a^{16} + \frac{423704442882707919203487020058565667646309316477554851481521459647810444171762626251930355363708}{38302460064194734332856012656541815136528577022762039872092298880794076157212570123194303598557681} a^{15} - \frac{325115763421636574500702395881599880049948461416180050970594108192089553699702354783732370440301}{38302460064194734332856012656541815136528577022762039872092298880794076157212570123194303598557681} a^{14} - \frac{497354051947843466924533498668798304134911247328492055296412937820836036242579301457214377933186}{38302460064194734332856012656541815136528577022762039872092298880794076157212570123194303598557681} a^{13} + \frac{272742112042304101060467444925907570858756201368316188001753692639736083955976832126423879496}{38302460064194734332856012656541815136528577022762039872092298880794076157212570123194303598557681} a^{12} + \frac{246315848234054484597462947746169943172065099918919539619861992073891894555519440784179971344811}{38302460064194734332856012656541815136528577022762039872092298880794076157212570123194303598557681} a^{11} + \frac{549827437673739211447630420339026340614805748903180292884354810396224400173849721739280844343254}{38302460064194734332856012656541815136528577022762039872092298880794076157212570123194303598557681} a^{10} - \frac{879454565233969923396032293376069477362466637554997364992141814065900445116404099604151220500966}{38302460064194734332856012656541815136528577022762039872092298880794076157212570123194303598557681} a^{9} - \frac{946522579662429329748165685689392681017234162358360830468104767220118618219944869744579957349812}{2015918950747091280676632245081148165080451422250633677478542046357582955642766848589173873608299} a^{8} - \frac{756488210675392253146419126764691890576034401645272283733367276242299610488481328305023045853633}{2015918950747091280676632245081148165080451422250633677478542046357582955642766848589173873608299} a^{7} + \frac{829242500717682063456178960883740291516022473727132957164938848819599560555727543952832916131048}{2015918950747091280676632245081148165080451422250633677478542046357582955642766848589173873608299} a^{6} - \frac{3836271754588694220738305334361609468219414099996333933092055831645849554232389525100415920868}{183265359158826480061512022280104378643677402022784879770776549668871177785706077144470352146209} a^{5} + \frac{624214598283967697247023325098359872863533965023407493633186346982998828749036180199713717569032}{2015918950747091280676632245081148165080451422250633677478542046357582955642766848589173873608299} a^{4} - \frac{79623379086195035222430507117012399569520025993516242611879597680487401526514092943809336990381}{2015918950747091280676632245081148165080451422250633677478542046357582955642766848589173873608299} a^{3} + \frac{23029071115923343199892112143406140748364232894296109540270893489001682881695755123993769470093}{183265359158826480061512022280104378643677402022784879770776549668871177785706077144470352146209} a^{2} - \frac{159526476502442748404854703102363806144796108527215311970133774350890022917573947450043429361040}{2015918950747091280676632245081148165080451422250633677478542046357582955642766848589173873608299} a - \frac{1201634167683977233094063782514502121136750419329175704091236079380789731368559447986528271359}{5319047363448789658777393786493794630819133040239138990708554212025284843384609099179878294481}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{9195132}$, which has order $147122112$ (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:  \( 15010229.973756868 \) (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

\(\Q(\sqrt{-133}) \), 3.3.361.1, 6.0.54355325248.1, 9.9.9025761726072081.1

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 $18$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ R $18$ $18$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ $18$

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$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19Data not computed