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Properties

Label 30.0.92509494307...6103.1
Degree $30$
Signature $[0, 15]$
Discriminant $-\,11^{27}\cdot 13^{25}$
Root discriminant $73.37$
Ramified primes $11, 13$
Class number $33050$ (GRH)
Class group $[33050]$ (GRH)
Galois group $C_{30}$ (as 30T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2728309, 2140857, 6118138, 3384997, 5324553, 3809737, 3605191, 1527807, 922095, -49818, 555027, 1044821, 549096, 909321, 605337, 108671, 37351, -31599, 547, -14696, 19150, 1088, 922, 324, -714, -98, -2, -21, 16, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 + 16*x^28 - 21*x^27 - 2*x^26 - 98*x^25 - 714*x^24 + 324*x^23 + 922*x^22 + 1088*x^21 + 19150*x^20 - 14696*x^19 + 547*x^18 - 31599*x^17 + 37351*x^16 + 108671*x^15 + 605337*x^14 + 909321*x^13 + 549096*x^12 + 1044821*x^11 + 555027*x^10 - 49818*x^9 + 922095*x^8 + 1527807*x^7 + 3605191*x^6 + 3809737*x^5 + 5324553*x^4 + 3384997*x^3 + 6118138*x^2 + 2140857*x + 2728309)
 
gp: K = bnfinit(x^30 - x^29 + 16*x^28 - 21*x^27 - 2*x^26 - 98*x^25 - 714*x^24 + 324*x^23 + 922*x^22 + 1088*x^21 + 19150*x^20 - 14696*x^19 + 547*x^18 - 31599*x^17 + 37351*x^16 + 108671*x^15 + 605337*x^14 + 909321*x^13 + 549096*x^12 + 1044821*x^11 + 555027*x^10 - 49818*x^9 + 922095*x^8 + 1527807*x^7 + 3605191*x^6 + 3809737*x^5 + 5324553*x^4 + 3384997*x^3 + 6118138*x^2 + 2140857*x + 2728309, 1)
 

Normalized defining polynomial

\( x^{30} - x^{29} + 16 x^{28} - 21 x^{27} - 2 x^{26} - 98 x^{25} - 714 x^{24} + 324 x^{23} + 922 x^{22} + 1088 x^{21} + 19150 x^{20} - 14696 x^{19} + 547 x^{18} - 31599 x^{17} + 37351 x^{16} + 108671 x^{15} + 605337 x^{14} + 909321 x^{13} + 549096 x^{12} + 1044821 x^{11} + 555027 x^{10} - 49818 x^{9} + 922095 x^{8} + 1527807 x^{7} + 3605191 x^{6} + 3809737 x^{5} + 5324553 x^{4} + 3384997 x^{3} + 6118138 x^{2} + 2140857 x + 2728309 \)

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

Invariants

Degree:  $30$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 15]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-92509494307745903779193517092075417558886189689936206103=-\,11^{27}\cdot 13^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.37$
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:  $11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(143=11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{143}(1,·)$, $\chi_{143}(3,·)$, $\chi_{143}(133,·)$, $\chi_{143}(134,·)$, $\chi_{143}(129,·)$, $\chi_{143}(9,·)$, $\chi_{143}(10,·)$, $\chi_{143}(140,·)$, $\chi_{143}(126,·)$, $\chi_{143}(14,·)$, $\chi_{143}(16,·)$, $\chi_{143}(17,·)$, $\chi_{143}(142,·)$, $\chi_{143}(90,·)$, $\chi_{143}(27,·)$, $\chi_{143}(92,·)$, $\chi_{143}(30,·)$, $\chi_{143}(95,·)$, $\chi_{143}(100,·)$, $\chi_{143}(101,·)$, $\chi_{143}(81,·)$, $\chi_{143}(42,·)$, $\chi_{143}(43,·)$, $\chi_{143}(48,·)$, $\chi_{143}(113,·)$, $\chi_{143}(51,·)$, $\chi_{143}(116,·)$, $\chi_{143}(53,·)$, $\chi_{143}(62,·)$, $\chi_{143}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{1039631} a^{25} - \frac{39058}{1039631} a^{24} - \frac{263107}{1039631} a^{23} + \frac{376187}{1039631} a^{22} - \frac{101039}{1039631} a^{21} + \frac{453782}{1039631} a^{20} + \frac{330184}{1039631} a^{19} - \frac{502932}{1039631} a^{18} + \frac{144917}{1039631} a^{17} + \frac{134876}{1039631} a^{16} - \frac{177666}{1039631} a^{15} - \frac{469932}{1039631} a^{14} + \frac{508462}{1039631} a^{13} + \frac{514088}{1039631} a^{12} + \frac{238075}{1039631} a^{11} - \frac{422387}{1039631} a^{10} + \frac{192052}{1039631} a^{9} - \frac{292430}{1039631} a^{8} + \frac{107116}{1039631} a^{7} + \frac{469260}{1039631} a^{6} + \frac{89202}{1039631} a^{5} + \frac{488035}{1039631} a^{4} - \frac{406311}{1039631} a^{3} - \frac{415042}{1039631} a^{2} + \frac{257272}{1039631} a - \frac{86620}{1039631}$, $\frac{1}{1039631} a^{26} + \frac{387837}{1039631} a^{24} - \frac{344215}{1039631} a^{23} - \frac{94116}{1039631} a^{22} + \frac{511796}{1039631} a^{21} + \frac{518252}{1039631} a^{20} + \frac{240816}{1039631} a^{19} + \frac{454606}{1039631} a^{18} - \frac{487733}{1039631} a^{17} - \frac{1135}{1039631} a^{16} - \frac{211635}{1039631} a^{15} - \frac{449920}{1039631} a^{14} - \frac{48109}{1039631} a^{13} + \frac{54045}{1039631} a^{12} - \frac{148701}{1039631} a^{11} + \frac{504945}{1039631} a^{10} - \frac{63079}{1039631} a^{9} - \frac{237658}{1039631} a^{8} - \frac{308787}{1039631} a^{7} - \frac{248248}{1039631} a^{6} - \frac{303361}{1039631} a^{5} - \frac{369666}{1039631} a^{4} - \frac{142865}{1039631} a^{3} + \frac{513019}{1039631} a^{2} + \frac{409541}{1039631} a - \frac{244686}{1039631}$, $\frac{1}{113319779} a^{27} - \frac{53}{113319779} a^{26} + \frac{40}{113319779} a^{25} + \frac{11520172}{113319779} a^{24} + \frac{38541945}{113319779} a^{23} - \frac{35094891}{113319779} a^{22} + \frac{39897772}{113319779} a^{21} - \frac{39779889}{113319779} a^{20} - \frac{3243758}{113319779} a^{19} - \frac{3044027}{113319779} a^{18} + \frac{28883094}{113319779} a^{17} - \frac{35131865}{113319779} a^{16} + \frac{49149252}{113319779} a^{15} + \frac{21980572}{113319779} a^{14} - \frac{32207862}{113319779} a^{13} - \frac{40504116}{113319779} a^{12} + \frac{45472994}{113319779} a^{11} + \frac{52496395}{113319779} a^{10} - \frac{39623208}{113319779} a^{9} - \frac{51642805}{113319779} a^{8} + \frac{55856225}{113319779} a^{7} + \frac{9124910}{113319779} a^{6} + \frac{53549083}{113319779} a^{5} - \frac{40772296}{113319779} a^{4} - \frac{15074511}{113319779} a^{3} - \frac{17933840}{113319779} a^{2} - \frac{29309014}{113319779} a - \frac{15642780}{113319779}$, $\frac{1}{34789172153} a^{28} - \frac{51}{34789172153} a^{27} + \frac{14976}{34789172153} a^{26} - \frac{9441}{34789172153} a^{25} + \frac{223384941}{34789172153} a^{24} + \frac{3222422936}{34789172153} a^{23} - \frac{2018663143}{34789172153} a^{22} - \frac{10144751088}{34789172153} a^{21} + \frac{14972256408}{34789172153} a^{20} - \frac{9701089200}{34789172153} a^{19} + \frac{5692929369}{34789172153} a^{18} - \frac{12814893760}{34789172153} a^{17} + \frac{15948355840}{34789172153} a^{16} - \frac{5830017735}{34789172153} a^{15} + \frac{12068237505}{34789172153} a^{14} + \frac{16697355168}{34789172153} a^{13} + \frac{11570334047}{34789172153} a^{12} + \frac{10158563815}{34789172153} a^{11} - \frac{196697592}{34789172153} a^{10} + \frac{2627710792}{34789172153} a^{9} + \frac{8640994514}{34789172153} a^{8} + \frac{6979036918}{34789172153} a^{7} - \frac{11792260426}{34789172153} a^{6} + \frac{1527757213}{34789172153} a^{5} + \frac{11794013881}{34789172153} a^{4} - \frac{3215507540}{34789172153} a^{3} - \frac{2621230291}{34789172153} a^{2} + \frac{7663889428}{34789172153} a + \frac{8259058}{113319779}$, $\frac{1}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{29} + \frac{76398869519579712048771673644937755389858303248250770077007415382541544634129}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{28} + \frac{18603055975127230079515155897818440318594676562825592465364796265987636417143158}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{27} + \frac{1400287237427087205714798511445567134978143717757274390061684347477567536628921088}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{26} + \frac{163734655921424060842504220711572123944810661887929524541440418433633507507320139}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{25} - \frac{2558018699921493537119508288603092927000108159040603833635006397156950279535407991261649}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{24} - \frac{972082304830941329728216573729321272138074438622376120871334382476642656123783047332376}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{23} + \frac{1693362599380622776884225705223782666989584026216418278473146629775517321665040467466598}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{22} + \frac{1139762739940787557563508829159105145211134869337732479221633873697064920396116042040123}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{21} + \frac{33642986963740406407442141551436905469099499724036934813319380908452781393046055540637}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{20} + \frac{2634511556467234376596435327766235107457124422425827947015317752315225863213166000475647}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{19} + \frac{1027420293353173243067064193503891906211448833079850505703916243785886814313730237666116}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{18} + \frac{1742197342864593537378368953116030466446292644871682631775887155790587179317037719082903}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{17} - \frac{2125257645537357877288979301806090462416199258891975992156225909871040224312950692278751}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{16} - \frac{19995339796091856107394709094200868374109589499701990048996106074154223915610613427486}{52546626865669426693082876479566165177981820518734985828197001269799255618456040222961} a^{15} - \frac{2262577720241680006609582245238717473018393741103806753763882163641775963916224098113089}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{14} + \frac{2672707098630796380113670135604388975086213312022510009786616551004364861326420347792168}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{13} - \frac{357879976073216458901474197595518843138062084666261317896031453376854737753480438353464}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{12} - \frac{2480123619774974697980069961317322701801788917743428388086869358188410185115698377986518}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{11} + \frac{77500309962382729051123519793530020983719940818699705391011754132435801192105717540752}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{10} + \frac{1460976861617994352931035985074451483381989366159422040084887373706861029293632788292077}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{9} - \frac{2000935784717457031778059395465465480136845123103060263817931034419825682356458205160553}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{8} + \frac{1213632020593658345174685131213649098254406951656251267407540503909302026894727731328984}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{7} - \frac{766925248072334157615333757827827342791600935215576564267132233791397621026156555861608}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{6} - \frac{1301731888246116515147438593810819376733734424929212886001930443941230785573259058335953}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{5} - \frac{1532607799243347795777346238650519672851411104559392364649269652818349622618021183938187}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{4} + \frac{2420954183295162747226187994790613703588504290191318239199069819621843940522619934601238}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{3} - \frac{1228819026668424359374933392455455387475947671298872796687652937380053315255976961339103}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a^{2} - \frac{2244429098495270240185171606566064712111654707350557329145260416255199152665031176950556}{5727582328357967509546033536272712004400018436542113455273473138408118862411708384302749} a - \frac{8041772616559397899380344479960031406790864770265457042129329177543032895792896368893}{18656619962078070063667861681670071675570092627173007997633462991557390431308496365807}$

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

Class group and class number

$C_{33050}$, which has order $33050$ (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:  $14$
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:  \( 85915831770.81862 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{30}$ (as 30T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 30
The 30 conjugacy class representatives for $C_{30}$
Character table for $C_{30}$ is not computed

Intermediate fields

\(\Q(\sqrt{-143}) \), 3.3.169.1, \(\Q(\zeta_{11})^+\), 6.0.494190983.1, 10.0.875489472034463.1, 15.15.432659002790862279847129.1

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 $15^{2}$ $15^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{3}$ $15^{2}$ R R $30$ $15^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{10}$ $30$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{3}$ $30$ $15^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{6}$ $30$

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
11Data not computed
13Data not computed