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Properties

Label 18.0.15267461241...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{16}\cdot 3^{53}\cdot 5^{16}\cdot 31^{12}$
Root discriminant $1940.83$
Ramified primes $2, 3, 5, 31$
Class number Not computed
Class group Not computed
Galois group $D_9:C_3$ (as 18T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12480520737495001, -1005446259, -4021784955, 18768329844, -14076247086, -14076248382, 18768331554, -4021787304, -1005443343, 223429361, 2907, -2304, 1554, -882, 414, -156, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 414*x^14 - 882*x^13 + 1554*x^12 - 2304*x^11 + 2907*x^10 + 223429361*x^9 - 1005443343*x^8 - 4021787304*x^7 + 18768331554*x^6 - 14076248382*x^5 - 14076247086*x^4 + 18768329844*x^3 - 4021784955*x^2 - 1005446259*x + 12480520737495001)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 414*x^14 - 882*x^13 + 1554*x^12 - 2304*x^11 + 2907*x^10 + 223429361*x^9 - 1005443343*x^8 - 4021787304*x^7 + 18768331554*x^6 - 14076248382*x^5 - 14076247086*x^4 + 18768329844*x^3 - 4021784955*x^2 - 1005446259*x + 12480520737495001, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 156 x^{15} + 414 x^{14} - 882 x^{13} + 1554 x^{12} - 2304 x^{11} + 2907 x^{10} + 223429361 x^{9} - 1005443343 x^{8} - 4021787304 x^{7} + 18768331554 x^{6} - 14076248382 x^{5} - 14076247086 x^{4} + 18768329844 x^{3} - 4021784955 x^{2} - 1005446259 x + 12480520737495001 \)

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:  \(-152674612414621913640596447375544095872332030000000000000000=-\,2^{16}\cdot 3^{53}\cdot 5^{16}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1940.83$
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:  $2, 3, 5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{155} a^{6} - \frac{3}{155} a^{5} + \frac{6}{155} a^{4} - \frac{7}{155} a^{3} + \frac{6}{155} a^{2} - \frac{3}{155} a + \frac{1}{155}$, $\frac{1}{155} a^{7} - \frac{3}{155} a^{5} + \frac{11}{155} a^{4} - \frac{3}{31} a^{3} + \frac{3}{31} a^{2} - \frac{8}{155} a + \frac{3}{155}$, $\frac{1}{155} a^{8} + \frac{2}{155} a^{5} + \frac{3}{155} a^{4} - \frac{6}{155} a^{3} + \frac{2}{31} a^{2} - \frac{6}{155} a + \frac{3}{155}$, $\frac{1}{4185} a^{9} + \frac{1}{465} a^{8} + \frac{1}{465} a^{7} + \frac{1}{1395} a^{6} - \frac{1}{465} a^{5} + \frac{14}{465} a^{4} - \frac{2}{45} a^{3} + \frac{5}{93} a^{2} - \frac{14}{465} a + \frac{383}{837}$, $\frac{1}{20925} a^{10} - \frac{1}{465} a^{8} + \frac{2}{1395} a^{7} + \frac{1}{465} a^{6} - \frac{34}{2325} a^{5} + \frac{29}{1395} a^{4} - \frac{2}{93} a^{3} + \frac{1}{93} a^{2} - \frac{481}{4185} a - \frac{1}{2325}$, $\frac{1}{20925} a^{11} + \frac{2}{1395} a^{8} + \frac{1}{465} a^{7} - \frac{4}{2325} a^{6} + \frac{2}{1395} a^{5} + \frac{8}{465} a^{4} - \frac{13}{465} a^{3} - \frac{319}{4185} a^{2} - \frac{46}{2325} a + \frac{4}{465}$, $\frac{1}{648675} a^{12} - \frac{2}{216225} a^{11} - \frac{2}{129735} a^{10} - \frac{2}{25947} a^{9} + \frac{11}{4805} a^{8} - \frac{352}{216225} a^{7} - \frac{418}{216225} a^{6} + \frac{208}{14415} a^{5} - \frac{893}{43245} a^{4} + \frac{6322}{25947} a^{3} - \frac{74393}{216225} a^{2} - \frac{141676}{648675} a + \frac{28886}{129735}$, $\frac{1}{648675} a^{13} - \frac{1}{43245} a^{11} + \frac{14}{648675} a^{10} - \frac{11}{129735} a^{9} + \frac{46}{72075} a^{8} - \frac{8}{4805} a^{7} - \frac{107}{43245} a^{6} + \frac{949}{72075} a^{5} + \frac{24929}{129735} a^{4} + \frac{257}{6975} a^{3} - \frac{15802}{43245} a^{2} + \frac{59761}{129735} a - \frac{147616}{648675}$, $\frac{1}{3243375} a^{14} - \frac{2}{3243375} a^{13} - \frac{2}{3243375} a^{12} + \frac{28}{3243375} a^{11} + \frac{22}{1081125} a^{10} + \frac{184}{3243375} a^{9} - \frac{2416}{1081125} a^{8} + \frac{1654}{1081125} a^{7} + \frac{2234}{1081125} a^{6} + \frac{1394254}{3243375} a^{5} + \frac{427711}{3243375} a^{4} + \frac{290653}{3243375} a^{3} + \frac{395248}{3243375} a^{2} - \frac{1378}{360375} a + \frac{1440106}{3243375}$, $\frac{1}{3243375} a^{15} - \frac{1}{3243375} a^{13} - \frac{1}{3243375} a^{12} + \frac{14}{1081125} a^{11} + \frac{16}{3243375} a^{10} + \frac{59}{648675} a^{9} - \frac{821}{360375} a^{8} + \frac{188}{120125} a^{7} + \frac{4363}{3243375} a^{6} + \frac{38547}{120125} a^{5} - \frac{12274}{129735} a^{4} + \frac{1027559}{3243375} a^{3} + \frac{6998}{1081125} a^{2} - \frac{726203}{3243375} a - \frac{1084838}{3243375}$, $\frac{1}{3243375} a^{16} + \frac{2}{3243375} a^{13} + \frac{2}{120125} a^{11} + \frac{56}{3243375} a^{10} - \frac{11}{648675} a^{9} - \frac{51}{120125} a^{8} + \frac{398}{648675} a^{7} + \frac{3137}{1081125} a^{6} + \frac{38667}{120125} a^{5} - \frac{201611}{648675} a^{4} - \frac{388816}{1081125} a^{3} + \frac{8523}{24025} a^{2} + \frac{281494}{648675} a + \frac{731126}{3243375}$, $\frac{1}{323833433421248912065254796399047418506870647259531920765709463922625} a^{17} + \frac{12852621390433151016687810469663718776401225761694187957302431}{323833433421248912065254796399047418506870647259531920765709463922625} a^{16} + \frac{561783832121522619309477803216105643112949001242272773081002}{64766686684249782413050959279809483701374129451906384153141892784525} a^{15} + \frac{1117345451250871382252397285137937092418276100957015047121954}{19049025495367583062662046847002789323933567485854818868571144936625} a^{14} - \frac{6155802379456087346022999415588102805030838906438271333849171}{64766686684249782413050959279809483701374129451906384153141892784525} a^{13} + \frac{26282745392608134743376545959683193436673817242055334930960477}{323833433421248912065254796399047418506870647259531920765709463922625} a^{12} - \frac{222693279096616784352594814016444388274268947788525113465941937}{323833433421248912065254796399047418506870647259531920765709463922625} a^{11} - \frac{6092633028230450939866702601073629130566292891695804210084150338}{323833433421248912065254796399047418506870647259531920765709463922625} a^{10} - \frac{11950624236702806049480151425532266114764144079571011841841716013}{323833433421248912065254796399047418506870647259531920765709463922625} a^{9} + \frac{89983482640688938647980611312742373005588257109487536037943742804}{64766686684249782413050959279809483701374129451906384153141892784525} a^{8} + \frac{997865387947956821011435676383743503541214459399039499328068272313}{323833433421248912065254796399047418506870647259531920765709463922625} a^{7} + \frac{14282534717368278532993757066467029864589805705794486703736781316}{19049025495367583062662046847002789323933567485854818868571144936625} a^{6} + \frac{56495206969547255128667153067897091700995331819124624728978227249913}{323833433421248912065254796399047418506870647259531920765709463922625} a^{5} - \frac{53969647101463476169975102501983076874574713387888568335328000836027}{323833433421248912065254796399047418506870647259531920765709463922625} a^{4} - \frac{7486605091905225331234756589407082706954760094328893541156221264999}{64766686684249782413050959279809483701374129451906384153141892784525} a^{3} + \frac{28108744958685139043991598845256148580014728215848770009135310430298}{323833433421248912065254796399047418506870647259531920765709463922625} a^{2} + \frac{56797193417306550675142394856251943264848997761522645134597662058424}{323833433421248912065254796399047418506870647259531920765709463922625} a + \frac{1367455071084800467933735866881453447069243859649476498858712}{2898713754915109996532687051940611742394315105145552375953875}$

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

Class group and class number

Not computed

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{5925950958515277866446857553593475753708332530}{33281528595084740644673608653439438702675520330882894183343941} a^{17} + \frac{1359996400408431558439762877443212803767198494385}{898601272067287997406187433642864844972239048933838142950286407} a^{16} - \frac{7135955387950082416615519551210081765702350420048}{898601272067287997406187433642864844972239048933838142950286407} a^{15} + \frac{1548218520563784019586833444614782256117429120230}{52858898356899293965069849037815579116014061701990478997075671} a^{14} - \frac{79038867792346764572348975095909293483093848963390}{898601272067287997406187433642864844972239048933838142950286407} a^{13} + \frac{197076284580355823957948931039126081311425516413340}{898601272067287997406187433642864844972239048933838142950286407} a^{12} - \frac{443878245288797937588533171174210476169141294180630}{898601272067287997406187433642864844972239048933838142950286407} a^{11} - \frac{297613796745441071904493032646294779692823317982308906}{299533757355762665802062477880954948324079682977946047650095469} a^{10} + \frac{1488967000663010104450182671009076671683161946479655430}{299533757355762665802062477880954948324079682977946047650095469} a^{9} - \frac{19363159840492802357715014304673364879778414211092334570}{299533757355762665802062477880954948324079682977946047650095469} a^{8} + \frac{205550351876017986397657481580366988210753526706154404010}{898601272067287997406187433642864844972239048933838142950286407} a^{7} + \frac{22502130967937567278565994075945749434344429895882846790}{52858898356899293965069849037815579116014061701990478997075671} a^{6} - \frac{1848230898438781494071140103232803688238589150789023118564}{898601272067287997406187433642864844972239048933838142950286407} a^{5} + \frac{2350452955394315435520194926154905715036144209272187606810}{898601272067287997406187433642864844972239048933838142950286407} a^{4} - \frac{1389633850609231501562673190067372465830890113669537506490}{898601272067287997406187433642864844972239048933838142950286407} a^{3} + \frac{4181671665204536831443362293166823359506644712516405000080}{898601272067287997406187433642864844972239048933838142950286407} a^{2} - \frac{33282804423232168657433314310200901987175538165294502546890840}{299533757355762665802062477880954948324079682977946047650095469} a + \frac{496540149305693657395802886040261647265506352149983474}{893733766497894938615697244981111262926524100711620573} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9:C_3$ (as 18T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $D_9:C_3$
Character table for $D_9:C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.24300.2 x3, 6.0.1771470000.4, 9.1.225591527925010157904900000000.2 x3

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

Sibling fields

Degree 9 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 R ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
2Data not computed
3Data not computed
5Data not computed
$31$31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$