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Properties

Label 18.18.6035459746...2512.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{24}\cdot 3^{24}\cdot 7^{4}\cdot 23\cdot 107^{6}\cdot 3520987^{4}$
Root discriminant $2705.59$
Ramified primes $2, 3, 7, 23, 107, 3520987$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 18T903

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11952636410206805283781, 9136064088352588936944, 2901079188765510844557, 474010864334666524928, 36294104440432333908, -226336313226170496, -266494192085567835, -16024771525133136, 319889009391855, 67489049880064, 1264320413073, -117303932784, -4525454337, 96122064, 6027618, -30736, -3852, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3852*x^16 - 30736*x^15 + 6027618*x^14 + 96122064*x^13 - 4525454337*x^12 - 117303932784*x^11 + 1264320413073*x^10 + 67489049880064*x^9 + 319889009391855*x^8 - 16024771525133136*x^7 - 266494192085567835*x^6 - 226336313226170496*x^5 + 36294104440432333908*x^4 + 474010864334666524928*x^3 + 2901079188765510844557*x^2 + 9136064088352588936944*x + 11952636410206805283781)
 
gp: K = bnfinit(x^18 - 3852*x^16 - 30736*x^15 + 6027618*x^14 + 96122064*x^13 - 4525454337*x^12 - 117303932784*x^11 + 1264320413073*x^10 + 67489049880064*x^9 + 319889009391855*x^8 - 16024771525133136*x^7 - 266494192085567835*x^6 - 226336313226170496*x^5 + 36294104440432333908*x^4 + 474010864334666524928*x^3 + 2901079188765510844557*x^2 + 9136064088352588936944*x + 11952636410206805283781, 1)
 

Normalized defining polynomial

\( x^{18} - 3852 x^{16} - 30736 x^{15} + 6027618 x^{14} + 96122064 x^{13} - 4525454337 x^{12} - 117303932784 x^{11} + 1264320413073 x^{10} + 67489049880064 x^{9} + 319889009391855 x^{8} - 16024771525133136 x^{7} - 266494192085567835 x^{6} - 226336313226170496 x^{5} + 36294104440432333908 x^{4} + 474010864334666524928 x^{3} + 2901079188765510844557 x^{2} + 9136064088352588936944 x + 11952636410206805283781 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(60354597469050827144261400747621547431798707274236593031872512=2^{24}\cdot 3^{24}\cdot 7^{4}\cdot 23\cdot 107^{6}\cdot 3520987^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2705.59$
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:  $2, 3, 7, 23, 107, 3520987$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{12} a^{12} + \frac{1}{4} a^{10} - \frac{1}{3} a^{9} - \frac{1}{4} a^{8} - \frac{1}{12} a^{6} - \frac{1}{4} a^{4} - \frac{1}{3} a^{3} + \frac{1}{4} a^{2} + \frac{1}{12}$, $\frac{1}{12} a^{13} + \frac{1}{4} a^{11} - \frac{1}{3} a^{10} - \frac{1}{4} a^{9} - \frac{1}{12} a^{7} - \frac{1}{4} a^{5} - \frac{1}{3} a^{4} + \frac{1}{4} a^{3} + \frac{1}{12} a$, $\frac{1}{36} a^{14} + \frac{1}{36} a^{12} - \frac{1}{9} a^{11} + \frac{5}{12} a^{10} + \frac{2}{9} a^{9} + \frac{17}{36} a^{8} - \frac{1}{3} a^{7} + \frac{11}{36} a^{6} - \frac{1}{9} a^{5} - \frac{1}{12} a^{4} - \frac{1}{9} a^{3} - \frac{17}{36} a^{2} + \frac{1}{3} a - \frac{7}{18}$, $\frac{1}{72} a^{15} - \frac{1}{72} a^{14} + \frac{1}{72} a^{13} - \frac{1}{36} a^{12} + \frac{19}{72} a^{11} - \frac{17}{36} a^{10} - \frac{1}{24} a^{9} - \frac{1}{36} a^{8} - \frac{13}{72} a^{7} + \frac{1}{4} a^{6} + \frac{1}{72} a^{5} + \frac{13}{36} a^{4} - \frac{25}{72} a^{3} + \frac{1}{36} a^{2} + \frac{5}{36} a - \frac{19}{72}$, $\frac{1}{72} a^{16} - \frac{1}{72} a^{13} - \frac{1}{72} a^{12} - \frac{5}{24} a^{11} - \frac{19}{72} a^{10} - \frac{5}{72} a^{9} - \frac{11}{24} a^{8} + \frac{5}{72} a^{7} - \frac{35}{72} a^{6} + \frac{3}{8} a^{5} - \frac{17}{72} a^{4} - \frac{23}{72} a^{3} + \frac{5}{12} a^{2} - \frac{1}{8} a + \frac{35}{72}$, $\frac{1}{530149610053730772423597962397972370455999615243823731936099486562236171202909977736431976410733483267937526758645962074833448} a^{17} + \frac{665721393534678420850732419057509656035321854794026937762250055662270317647076857117665922603113879870228981816538694999243}{265074805026865386211798981198986185227999807621911865968049743281118085601454988868215988205366741633968763379322981037416724} a^{16} + \frac{265352769385800336946029011145475874772072276672310458802478360647051936949791759053296226556967345971740163939876145876443}{58905512228192308047066440266441374495111068360424859104011054062470685689212219748492441823414831474215280750960662452759272} a^{15} + \frac{1486581398615597334384240609306093240850487357931764308517598738897347481090180642429316550976581512796324190980301942024389}{132537402513432693105899490599493092613999903810955932984024871640559042800727494434107994102683370816984381689661490518708362} a^{14} - \frac{165984460824578611079148133576610318147254080370539693045738237395984797634225748762012845834124730370874903904167921345905}{14726378057048077011766610066610343623777767090106214776002763515617671422303054937123110455853707868553820187740165613189818} a^{13} + \frac{13875204518069879805019763828642099081752384008841478674034464132908230321135245523964096995378432713803242657607462108156289}{530149610053730772423597962397972370455999615243823731936099486562236171202909977736431976410733483267937526758645962074833448} a^{12} - \frac{47180560156855926318414143222323060174063519075869934543126690223040147939767344868620175562114498860085328702212667994999347}{132537402513432693105899490599493092613999903810955932984024871640559042800727494434107994102683370816984381689661490518708362} a^{11} + \frac{80317512588558371563452642344036195645710118319145473653962272326014662383289392430545979509943268313501690377646846268986615}{530149610053730772423597962397972370455999615243823731936099486562236171202909977736431976410733483267937526758645962074833448} a^{10} + \frac{32166874068244689848740122035998601345426344093044997115880837415445390522211953427014503178921469575332433297559309898614355}{66268701256716346552949745299746546306999951905477966492012435820279521400363747217053997051341685408492190844830745259354181} a^{9} - \frac{9754636596365760046267633509581860135413322328610690115553681481615832544035468982064864406624009579198957440613942905805411}{530149610053730772423597962397972370455999615243823731936099486562236171202909977736431976410733483267937526758645962074833448} a^{8} + \frac{29085120169635844159312529417449314941790186064645648203437053005068664588756184072235487924977759116679288172009488693682905}{132537402513432693105899490599493092613999903810955932984024871640559042800727494434107994102683370816984381689661490518708362} a^{7} + \frac{86013356551096501095814415473156379450554093749655284512897748776222954789942959076307606802630904463008786379446218827732659}{530149610053730772423597962397972370455999615243823731936099486562236171202909977736431976410733483267937526758645962074833448} a^{6} - \frac{7813344102753271226849817804605904023986283291857245901876827691447641759712311469450931915220797914830358263811400508151979}{132537402513432693105899490599493092613999903810955932984024871640559042800727494434107994102683370816984381689661490518708362} a^{5} + \frac{19424613154712790986716904916050324245829140713700025824285456607020998672304407168540223774564576087780304318100186114154279}{176716536684576924141199320799324123485333205081274577312033162187412057067636659245477325470244494422645842252881987358277816} a^{4} + \frac{44450385305621373193325736314728495391525652618085262684326631093075179463880102739977697154805894362743399115456890984622033}{530149610053730772423597962397972370455999615243823731936099486562236171202909977736431976410733483267937526758645962074833448} a^{3} + \frac{17701724048728286004774003042051089504936943536316950737696119925816060195038474929336110459607218644972925608958439098697893}{58905512228192308047066440266441374495111068360424859104011054062470685689212219748492441823414831474215280750960662452759272} a^{2} + \frac{127944165122136871385382759825581649601455007970707608487331686279326344184203538698072127865759529585820591318748474958460155}{530149610053730772423597962397972370455999615243823731936099486562236171202909977736431976410733483267937526758645962074833448} a - \frac{65051286762198755695268279156264835515150758816718679682300782712270592293733413431160427324169644725029686739193211170656107}{530149610053730772423597962397972370455999615243823731936099486562236171202909977736431976410733483267937526758645962074833448}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $17$
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:  \( 7991505074050000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T903:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 559872
The 174 conjugacy class representatives for t18n903 are not computed
Character table for t18n903 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 3.3.321.1, 6.6.19783872.1

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

Sibling fields

Degree 18 sibling: data not computed

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 ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ R ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.12.18.64$x^{12} + 14 x^{11} + 12 x^{10} + 4 x^{9} + 10 x^{8} + 4 x^{6} + 8 x^{5} + 8 x^{4} + 16 x - 8$$4$$3$$18$12T51$[2, 2, 2, 2]^{6}$
$3$3.6.6.7$x^{6} + 3 x + 6$$6$$1$$6$$C_3^2:D_4$$[5/4, 5/4]_{4}^{2}$
3.12.18.12$x^{12} + 18 x^{11} + 33 x^{10} - 3 x^{9} - 36 x^{8} - 27 x^{7} + 21 x^{6} - 27 x^{5} - 18 x^{4} + 9 x^{3} - 27 x + 36$$6$$2$$18$12T171$[3/2, 3/2, 2, 2]_{2}^{4}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
107Data not computed
3520987Data not computed