↑

Properties

Label 18.0.25017305116...0063.6
Degree $18$
Signature $[0, 9]$
Discriminant $-\,19^{17}\cdot 37^{17}$
Root discriminant $488.42$
Ramified primes $19, 37$
Class number $7801759098$ (GRH)
Class group $[7801759098]$ (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![11080414060544, 355897168384, -1463520329344, 136367346344, 171707544098, -44238428233, -3711440771, 2705060950, -87377712, -62092191, 5297847, 565724, 156314, -44727, 4523, -58, 20, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 20*x^16 - 58*x^15 + 4523*x^14 - 44727*x^13 + 156314*x^12 + 565724*x^11 + 5297847*x^10 - 62092191*x^9 - 87377712*x^8 + 2705060950*x^7 - 3711440771*x^6 - 44238428233*x^5 + 171707544098*x^4 + 136367346344*x^3 - 1463520329344*x^2 + 355897168384*x + 11080414060544)
 
gp: K = bnfinit(x^18 - x^17 + 20*x^16 - 58*x^15 + 4523*x^14 - 44727*x^13 + 156314*x^12 + 565724*x^11 + 5297847*x^10 - 62092191*x^9 - 87377712*x^8 + 2705060950*x^7 - 3711440771*x^6 - 44238428233*x^5 + 171707544098*x^4 + 136367346344*x^3 - 1463520329344*x^2 + 355897168384*x + 11080414060544, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 20 x^{16} - 58 x^{15} + 4523 x^{14} - 44727 x^{13} + 156314 x^{12} + 565724 x^{11} + 5297847 x^{10} - 62092191 x^{9} - 87377712 x^{8} + 2705060950 x^{7} - 3711440771 x^{6} - 44238428233 x^{5} + 171707544098 x^{4} + 136367346344 x^{3} - 1463520329344 x^{2} + 355897168384 x + 11080414060544 \)

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:  \(-2501730511640249522911904558507505938447107600063=-\,19^{17}\cdot 37^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $488.42$
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:  $19, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(703=19\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{703}(1,·)$, $\chi_{703}(581,·)$, $\chi_{703}(582,·)$, $\chi_{703}(136,·)$, $\chi_{703}(585,·)$, $\chi_{703}(336,·)$, $\chi_{703}(280,·)$, $\chi_{703}(218,·)$, $\chi_{703}(287,·)$, $\chi_{703}(416,·)$, $\chi_{703}(485,·)$, $\chi_{703}(423,·)$, $\chi_{703}(367,·)$, $\chi_{703}(118,·)$, $\chi_{703}(567,·)$, $\chi_{703}(121,·)$, $\chi_{703}(122,·)$, $\chi_{703}(702,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{64} a^{7} - \frac{1}{32} a^{6} + \frac{1}{32} a^{5} + \frac{5}{64} a^{3} + \frac{1}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{1408} a^{8} - \frac{1}{352} a^{7} + \frac{13}{704} a^{6} + \frac{1}{44} a^{5} - \frac{5}{128} a^{4} - \frac{19}{352} a^{3} + \frac{87}{352} a^{2} - \frac{41}{88} a + \frac{3}{11}$, $\frac{1}{2816} a^{9} - \frac{1}{2816} a^{8} - \frac{1}{352} a^{7} + \frac{3}{128} a^{6} - \frac{91}{2816} a^{5} + \frac{23}{2816} a^{4} + \frac{49}{1408} a^{3} + \frac{21}{352} a^{2} - \frac{1}{11}$, $\frac{1}{11264} a^{10} - \frac{1}{5632} a^{9} + \frac{1}{11264} a^{8} - \frac{23}{5632} a^{7} + \frac{227}{11264} a^{6} + \frac{97}{5632} a^{5} - \frac{365}{11264} a^{4} + \frac{215}{5632} a^{3} - \frac{47}{1408} a^{2} + \frac{79}{176} a - \frac{5}{11}$, $\frac{1}{22528} a^{11} - \frac{1}{22528} a^{10} - \frac{1}{22528} a^{9} + \frac{3}{22528} a^{8} + \frac{15}{2048} a^{7} + \frac{613}{22528} a^{6} + \frac{1013}{22528} a^{5} - \frac{463}{22528} a^{4} - \frac{1005}{11264} a^{3} - \frac{25}{256} a^{2} - \frac{163}{352} a - \frac{9}{22}$, $\frac{1}{22528} a^{12} - \frac{1}{11264} a^{9} - \frac{3}{11264} a^{8} - \frac{9}{11264} a^{7} + \frac{5}{352} a^{6} + \frac{469}{11264} a^{5} + \frac{845}{22528} a^{4} - \frac{2731}{11264} a^{3} - \frac{529}{2816} a^{2} + \frac{71}{352} a + \frac{3}{22}$, $\frac{1}{3964928} a^{13} - \frac{39}{1982464} a^{12} - \frac{3}{991232} a^{11} - \frac{5}{495616} a^{10} + \frac{203}{1982464} a^{9} + \frac{115}{991232} a^{8} + \frac{4911}{991232} a^{7} + \frac{10707}{495616} a^{6} - \frac{158735}{3964928} a^{5} - \frac{567}{1982464} a^{4} + \frac{32121}{495616} a^{3} + \frac{4777}{30976} a^{2} - \frac{21}{704} a - \frac{85}{484}$, $\frac{1}{15859712} a^{14} + \frac{1}{15859712} a^{13} - \frac{95}{7929856} a^{12} - \frac{71}{3964928} a^{11} + \frac{31}{7929856} a^{10} + \frac{779}{7929856} a^{9} + \frac{199}{991232} a^{8} + \frac{9159}{3964928} a^{7} - \frac{170023}{15859712} a^{6} + \frac{21363}{1441792} a^{5} - \frac{243493}{7929856} a^{4} + \frac{1535}{1982464} a^{3} + \frac{4021}{123904} a^{2} + \frac{7187}{30976} a - \frac{467}{1936}$, $\frac{1}{31719424} a^{15} + \frac{1}{31719424} a^{13} - \frac{13}{1441792} a^{12} - \frac{25}{1441792} a^{11} - \frac{69}{3964928} a^{10} + \frac{365}{15859712} a^{9} - \frac{181}{7929856} a^{8} - \frac{171843}{31719424} a^{7} - \frac{7741}{3964928} a^{6} - \frac{110459}{31719424} a^{5} + \frac{608577}{15859712} a^{4} + \frac{213121}{3964928} a^{3} + \frac{25611}{247808} a^{2} + \frac{12709}{61952} a - \frac{1509}{3872}$, $\frac{1}{111856138059776} a^{16} + \frac{455}{84039172096} a^{15} - \frac{1519831}{111856138059776} a^{14} - \frac{8249425}{111856138059776} a^{13} + \frac{351096949}{27964034514944} a^{12} - \frac{203799811}{55928069029888} a^{11} + \frac{2480186081}{55928069029888} a^{10} + \frac{7854355215}{55928069029888} a^{9} + \frac{35395995289}{111856138059776} a^{8} + \frac{251655344713}{111856138059776} a^{7} - \frac{1002211057691}{111856138059776} a^{6} + \frac{6820711875299}{111856138059776} a^{5} - \frac{223395640957}{5084369911808} a^{4} - \frac{1766528381971}{13982017257472} a^{3} + \frac{201152939555}{873876078592} a^{2} + \frac{29003819041}{218469019648} a + \frac{6000677135}{13654313728}$, $\frac{1}{1529887672445755364734212719346637387362992128} a^{17} + \frac{1074866826441302769073883098929}{764943836222877682367106359673318693681496064} a^{16} + \frac{4481450476002649527575127166853242365}{764943836222877682367106359673318693681496064} a^{15} + \frac{73009700038568266214336738783342391}{34770174373767167380323016348787213349158912} a^{14} + \frac{182956077985252063689778691313859923335}{1529887672445755364734212719346637387362992128} a^{13} - \frac{74906717758841139005191924099015394787}{69540348747534334760646032697574426698317824} a^{12} - \frac{6454595390164067877459943807891099970475}{382471918111438841183553179836659346840748032} a^{11} + \frac{6660876701654867480246131844104068875691}{191235959055719420591776589918329673420374016} a^{10} + \frac{142826596797572559155691497690982144651199}{1529887672445755364734212719346637387362992128} a^{9} - \frac{17552600044164428841276797445640512729027}{69540348747534334760646032697574426698317824} a^{8} + \frac{1634451078023403462617586708089982960349669}{764943836222877682367106359673318693681496064} a^{7} + \frac{7281083633784992057788426409535387468068317}{382471918111438841183553179836659346840748032} a^{6} + \frac{51156437352824181338897321330496525388370521}{1529887672445755364734212719346637387362992128} a^{5} - \frac{42521790194251318053891698461230892502497519}{764943836222877682367106359673318693681496064} a^{4} - \frac{33159951066683104500089059071003124793094999}{191235959055719420591776589918329673420374016} a^{3} - \frac{99954356271476870195642874429171310187941}{11952247440982463786986036869895604588773376} a^{2} - \frac{1208037248344304174274867070616620996498227}{2988061860245615946746509217473901147193344} a - \frac{21014965218539127052296888300703128501101}{186753866265350996671656826092118821699584}$

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

Class group and class number

$C_{7801759098}$, which has order $7801759098$ (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:  \( 4738886400565.654 \) (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

\(\Q(\sqrt{-703}) \), 3.3.494209.2, 6.0.171702502583743.1, 9.9.59654416235884558133761.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{18}$ $18$ $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ R $18$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
19Data not computed
37Data not computed