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Properties

Label 30.0.49280433368...7971.1
Degree $30$
Signature $[0, 15]$
Discriminant $-\,11^{27}\cdot 19^{20}$
Root discriminant $61.62$
Ramified primes $11, 19$
Class number $3904$ (GRH)
Class group $[2, 2, 2, 2, 2, 122]$ (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![282475249, -242121642, 247886443, -206709293, 178002937, -146691496, 121634660, -99785217, 81950638, -67122055, 54985393, -13825603, 10116708, -2791494, 1862867, -550284, 339011, -103069, 58163, -16148, 7426, -363, 1309, -113, 232, -28, 41, -6, 7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 + 7*x^28 - 6*x^27 + 41*x^26 - 28*x^25 + 232*x^24 - 113*x^23 + 1309*x^22 - 363*x^21 + 7426*x^20 - 16148*x^19 + 58163*x^18 - 103069*x^17 + 339011*x^16 - 550284*x^15 + 1862867*x^14 - 2791494*x^13 + 10116708*x^12 - 13825603*x^11 + 54985393*x^10 - 67122055*x^9 + 81950638*x^8 - 99785217*x^7 + 121634660*x^6 - 146691496*x^5 + 178002937*x^4 - 206709293*x^3 + 247886443*x^2 - 242121642*x + 282475249)
 
gp: K = bnfinit(x^30 - x^29 + 7*x^28 - 6*x^27 + 41*x^26 - 28*x^25 + 232*x^24 - 113*x^23 + 1309*x^22 - 363*x^21 + 7426*x^20 - 16148*x^19 + 58163*x^18 - 103069*x^17 + 339011*x^16 - 550284*x^15 + 1862867*x^14 - 2791494*x^13 + 10116708*x^12 - 13825603*x^11 + 54985393*x^10 - 67122055*x^9 + 81950638*x^8 - 99785217*x^7 + 121634660*x^6 - 146691496*x^5 + 178002937*x^4 - 206709293*x^3 + 247886443*x^2 - 242121642*x + 282475249, 1)
 

Normalized defining polynomial

\( x^{30} - x^{29} + 7 x^{28} - 6 x^{27} + 41 x^{26} - 28 x^{25} + 232 x^{24} - 113 x^{23} + 1309 x^{22} - 363 x^{21} + 7426 x^{20} - 16148 x^{19} + 58163 x^{18} - 103069 x^{17} + 339011 x^{16} - 550284 x^{15} + 1862867 x^{14} - 2791494 x^{13} + 10116708 x^{12} - 13825603 x^{11} + 54985393 x^{10} - 67122055 x^{9} + 81950638 x^{8} - 99785217 x^{7} + 121634660 x^{6} - 146691496 x^{5} + 178002937 x^{4} - 206709293 x^{3} + 247886443 x^{2} - 242121642 x + 282475249 \)

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:  \(-492804333687064066258882287420865640909549691972387971=-\,11^{27}\cdot 19^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.62$
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:  $11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(209=11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{209}(64,·)$, $\chi_{209}(1,·)$, $\chi_{209}(68,·)$, $\chi_{209}(197,·)$, $\chi_{209}(134,·)$, $\chi_{209}(7,·)$, $\chi_{209}(201,·)$, $\chi_{209}(140,·)$, $\chi_{209}(144,·)$, $\chi_{209}(83,·)$, $\chi_{209}(20,·)$, $\chi_{209}(87,·)$, $\chi_{209}(153,·)$, $\chi_{209}(26,·)$, $\chi_{209}(30,·)$, $\chi_{209}(159,·)$, $\chi_{209}(96,·)$, $\chi_{209}(163,·)$, $\chi_{209}(102,·)$, $\chi_{209}(39,·)$, $\chi_{209}(106,·)$, $\chi_{209}(172,·)$, $\chi_{209}(45,·)$, $\chi_{209}(49,·)$, $\chi_{209}(178,·)$, $\chi_{209}(115,·)$, $\chi_{209}(182,·)$, $\chi_{209}(58,·)$, $\chi_{209}(125,·)$, $\chi_{209}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2687693557171} a^{21} - \frac{347993380701}{2687693557171} a^{20} + \frac{91335311428}{383956222453} a^{19} - \frac{39613907024}{2687693557171} a^{18} - \frac{1247950235078}{2687693557171} a^{17} + \frac{15758562652}{383956222453} a^{16} + \frac{187771973313}{2687693557171} a^{15} - \frac{198483315962}{2687693557171} a^{14} - \frac{84344118769}{383956222453} a^{13} + \frac{713912748802}{2687693557171} a^{12} - \frac{270361834492}{2687693557171} a^{11} - \frac{1325336643854}{2687693557171} a^{10} + \frac{151255507988}{383956222453} a^{9} + \frac{1217700603867}{2687693557171} a^{8} + \frac{1233061338993}{2687693557171} a^{7} + \frac{6599198538}{383956222453} a^{6} - \frac{250083471765}{2687693557171} a^{5} + \frac{1095598511799}{2687693557171} a^{4} + \frac{59279277592}{383956222453} a^{3} - \frac{967335289047}{2687693557171} a^{2} + \frac{375480301820}{2687693557171} a - \frac{83873410989}{383956222453}$, $\frac{1}{18813854900197} a^{22} - \frac{1}{18813854900197} a^{21} + \frac{104564229022}{2687693557171} a^{20} - \frac{1023303402449}{18813854900197} a^{19} + \frac{5415001021366}{18813854900197} a^{18} - \frac{918739173426}{2687693557171} a^{17} - \frac{5869653275359}{18813854900197} a^{16} + \frac{5187615141029}{18813854900197} a^{15} + \frac{1235074315704}{2687693557171} a^{14} + \frac{206452608930}{18813854900197} a^{13} - \frac{6089299863144}{18813854900197} a^{12} + \frac{6971085576981}{18813854900197} a^{11} - \frac{90489932509}{383956222453} a^{10} + \frac{7388248338050}{18813854900197} a^{9} + \frac{6845380067646}{18813854900197} a^{8} + \frac{920866158581}{2687693557171} a^{7} - \frac{7725318838826}{18813854900197} a^{6} + \frac{250083471765}{18813854900197} a^{5} - \frac{211364962036}{2687693557171} a^{4} + \frac{5344388393651}{18813854900197} a^{3} + \frac{6342722403389}{18813854900197} a^{2} - \frac{492447154492}{2687693557171} a + \frac{100049417038}{383956222453}$, $\frac{1}{131696984301379} a^{23} - \frac{1}{131696984301379} a^{22} + \frac{1}{18813854900197} a^{21} - \frac{32384539016241}{131696984301379} a^{20} - \frac{38143710432311}{131696984301379} a^{19} - \frac{3495219909101}{18813854900197} a^{18} - \frac{35996543439709}{131696984301379} a^{17} + \frac{17282287135427}{131696984301379} a^{16} - \frac{1348052916499}{18813854900197} a^{15} - \frac{7148726548529}{131696984301379} a^{14} - \frac{60190470297819}{131696984301379} a^{13} + \frac{3251076090137}{131696984301379} a^{12} - \frac{394776139033}{2687693557171} a^{11} + \frac{12608148172843}{131696984301379} a^{10} + \frac{41967579912182}{131696984301379} a^{9} - \frac{1086036730851}{18813854900197} a^{8} - \frac{47426179105187}{131696984301379} a^{7} + \frac{32191727191961}{131696984301379} a^{6} + \frac{3589478752765}{18813854900197} a^{5} - \frac{32262257552519}{131696984301379} a^{4} + \frac{13271502608239}{131696984301379} a^{3} - \frac{4422941291124}{18813854900197} a^{2} + \frac{361111099987}{2687693557171} a + \frac{61631266567}{383956222453}$, $\frac{1}{921878890109653} a^{24} - \frac{1}{921878890109653} a^{23} + \frac{1}{131696984301379} a^{22} - \frac{6}{921878890109653} a^{21} + \frac{410617380974198}{921878890109653} a^{20} - \frac{8227136317628}{131696984301379} a^{19} - \frac{244342430260417}{921878890109653} a^{18} + \frac{7487701410468}{921878890109653} a^{17} - \frac{4702026045348}{131696984301379} a^{16} + \frac{211201159176631}{921878890109653} a^{15} - \frac{356272343207971}{921878890109653} a^{14} - \frac{266973215880279}{921878890109653} a^{13} - \frac{8005157737370}{18813854900197} a^{12} - \frac{15977408167729}{921878890109653} a^{11} + \frac{403042972767261}{921878890109653} a^{10} + \frac{55150075430210}{131696984301379} a^{9} + \frac{226523135850675}{921878890109653} a^{8} + \frac{301686391040707}{921878890109653} a^{7} + \frac{10327222815003}{131696984301379} a^{6} - \frac{364025822944666}{921878890109653} a^{5} + \frac{143937248130782}{921878890109653} a^{4} + \frac{3099930336525}{131696984301379} a^{3} - \frac{7919250681460}{18813854900197} a^{2} - \frac{926987422094}{2687693557171} a - \frac{6101061167}{383956222453}$, $\frac{1}{6453152230767571} a^{25} - \frac{1}{6453152230767571} a^{24} + \frac{1}{921878890109653} a^{23} - \frac{6}{6453152230767571} a^{22} + \frac{41}{6453152230767571} a^{21} + \frac{158894261152623}{921878890109653} a^{20} - \frac{2548981149612435}{6453152230767571} a^{19} + \frac{2769387887255317}{6453152230767571} a^{18} + \frac{375549781869106}{921878890109653} a^{17} + \frac{2597763033928686}{6453152230767571} a^{16} + \frac{295281861523130}{6453152230767571} a^{15} - \frac{2076336518347965}{6453152230767571} a^{14} + \frac{54549229238466}{131696984301379} a^{13} - \frac{157653850575572}{6453152230767571} a^{12} + \frac{1660771618248821}{6453152230767571} a^{11} + \frac{456769492222349}{921878890109653} a^{10} + \frac{3078200313061068}{6453152230767571} a^{9} + \frac{697659872036758}{6453152230767571} a^{8} + \frac{204904820088475}{921878890109653} a^{7} - \frac{1513581240041585}{6453152230767571} a^{6} + \frac{2096898326479699}{6453152230767571} a^{5} - \frac{162578511770562}{921878890109653} a^{4} - \frac{3404287526023}{131696984301379} a^{3} + \frac{4558742416915}{18813854900197} a^{2} + \frac{989357852197}{2687693557171} a - \frac{36554956941}{383956222453}$, $\frac{1}{45172065615372997} a^{26} - \frac{1}{45172065615372997} a^{25} + \frac{1}{6453152230767571} a^{24} - \frac{6}{45172065615372997} a^{23} + \frac{41}{45172065615372997} a^{22} - \frac{4}{6453152230767571} a^{21} - \frac{20078461787858879}{45172065615372997} a^{20} + \frac{1936435451245882}{45172065615372997} a^{19} + \frac{1872712952531377}{6453152230767571} a^{18} - \frac{6523413629137509}{45172065615372997} a^{17} + \frac{8388274563430523}{45172065615372997} a^{16} + \frac{8464606163798751}{45172065615372997} a^{15} - \frac{77527636473024}{921878890109653} a^{14} - \frac{22211783732134648}{45172065615372997} a^{13} + \frac{13498836140283852}{45172065615372997} a^{12} + \frac{1046677802593288}{6453152230767571} a^{11} + \frac{8527917329353554}{45172065615372997} a^{10} + \frac{10594422595708862}{45172065615372997} a^{9} + \frac{2194046483572863}{6453152230767571} a^{8} + \frac{17559500263972015}{45172065615372997} a^{7} + \frac{13235213369931274}{45172065615372997} a^{6} + \frac{2705971921068445}{6453152230767571} a^{5} + \frac{119585457947593}{921878890109653} a^{4} + \frac{57578321788541}{131696984301379} a^{3} + \frac{5199994814393}{18813854900197} a^{2} + \frac{684989225999}{2687693557171} a + \frac{178121356982}{383956222453}$, $\frac{1}{316204459307610979} a^{27} - \frac{1}{316204459307610979} a^{26} + \frac{1}{45172065615372997} a^{25} - \frac{6}{316204459307610979} a^{24} + \frac{41}{316204459307610979} a^{23} - \frac{4}{45172065615372997} a^{22} + \frac{232}{316204459307610979} a^{21} - \frac{2639418730749424}{316204459307610979} a^{20} + \frac{2090329875050513}{45172065615372997} a^{19} - \frac{30468821509848511}{316204459307610979} a^{18} + \frac{99786745146724089}{316204459307610979} a^{17} - \frac{35254820038831175}{316204459307610979} a^{16} + \frac{2132430225359697}{6453152230767571} a^{15} + \frac{66284755443845441}{316204459307610979} a^{14} - \frac{2338468767523727}{316204459307610979} a^{13} - \frac{19049609785924496}{45172065615372997} a^{12} - \frac{49099174611974761}{316204459307610979} a^{11} - \frac{134944799154298202}{316204459307610979} a^{10} + \frac{19948596799155608}{45172065615372997} a^{9} - \frac{99930835432580762}{316204459307610979} a^{8} - \frac{6841693082971116}{316204459307610979} a^{7} + \frac{9790494905500051}{45172065615372997} a^{6} - \frac{1697641831375776}{6453152230767571} a^{5} + \frac{379851092944625}{921878890109653} a^{4} - \frac{62332933334020}{131696984301379} a^{3} + \frac{1957476928989}{18813854900197} a^{2} - \frac{160180521682}{2687693557171} a + \frac{142341971271}{383956222453}$, $\frac{1}{2213431215153276853} a^{28} - \frac{1}{2213431215153276853} a^{27} + \frac{1}{316204459307610979} a^{26} - \frac{6}{2213431215153276853} a^{25} + \frac{41}{2213431215153276853} a^{24} - \frac{4}{316204459307610979} a^{23} + \frac{232}{2213431215153276853} a^{22} - \frac{113}{2213431215153276853} a^{21} + \frac{22998122927473237}{316204459307610979} a^{20} - \frac{608093417465014305}{2213431215153276853} a^{19} - \frac{639416634734387385}{2213431215153276853} a^{18} + \frac{159909599525664885}{2213431215153276853} a^{17} + \frac{12258398580122818}{45172065615372997} a^{16} + \frac{309742053893813239}{2213431215153276853} a^{15} - \frac{13268104964604258}{2213431215153276853} a^{14} - \frac{80563214878602899}{316204459307610979} a^{13} + \frac{439097036466010565}{2213431215153276853} a^{12} + \frac{511233634186991577}{2213431215153276853} a^{11} + \frac{55597467244400945}{316204459307610979} a^{10} - \frac{190042436674527599}{2213431215153276853} a^{9} - \frac{841993102306680617}{2213431215153276853} a^{8} + \frac{30369023154554925}{316204459307610979} a^{7} + \frac{2835937047508706}{45172065615372997} a^{6} - \frac{963706206054899}{6453152230767571} a^{5} + \frac{182827005441193}{921878890109653} a^{4} + \frac{45450265384868}{131696984301379} a^{3} - \frac{3773388389833}{18813854900197} a^{2} - \frac{915216749516}{2687693557171} a - \frac{171658316488}{383956222453}$, $\frac{1}{15494018506072937971} a^{29} - \frac{1}{15494018506072937971} a^{28} + \frac{1}{2213431215153276853} a^{27} - \frac{6}{15494018506072937971} a^{26} + \frac{41}{15494018506072937971} a^{25} - \frac{4}{2213431215153276853} a^{24} + \frac{232}{15494018506072937971} a^{23} - \frac{113}{15494018506072937971} a^{22} + \frac{187}{2213431215153276853} a^{21} - \frac{3479417198467931146}{15494018506072937971} a^{20} + \frac{709442320375857025}{15494018506072937971} a^{19} - \frac{1836350343822045197}{15494018506072937971} a^{18} - \frac{56508114635212096}{316204459307610979} a^{17} - \frac{3283108203175879303}{15494018506072937971} a^{16} + \frac{4803307105815082642}{15494018506072937971} a^{15} + \frac{371116553352958075}{2213431215153276853} a^{14} + \frac{3240269339188634206}{15494018506072937971} a^{13} - \frac{512279875877630475}{15494018506072937971} a^{12} + \frac{1021510001879786492}{2213431215153276853} a^{11} - \frac{3036382400176786823}{15494018506072937971} a^{10} - \frac{162773376821418747}{15494018506072937971} a^{9} + \frac{140332183717355470}{2213431215153276853} a^{8} - \frac{110462990294888496}{316204459307610979} a^{7} - \frac{15530011379721292}{45172065615372997} a^{6} - \frac{1990433042577867}{6453152230767571} a^{5} - \frac{184117649258759}{921878890109653} a^{4} - \frac{7575044230753}{131696984301379} a^{3} - \frac{5642386901759}{18813854900197} a^{2} - \frac{1191310653665}{2687693557171} a + \frac{92602423158}{383956222453}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{122}$, which has order $3904$ (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:  \( \frac{42015}{18813854900197} a^{24} - \frac{642095026}{18813854900197} a^{13} - \frac{12607356842541}{18813854900197} a^{2} \) (order $22$)
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:  \( 649367471590.597 \) (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

\(\Q(\sqrt{-11}) \), 3.3.361.1, \(\Q(\zeta_{11})^+\), 6.0.173457251.1, \(\Q(\zeta_{11})\), 15.15.19241912323039288533050521.1

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 $30$ $15^{2}$ $15^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{3}$ R $30$ $30$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{10}$ $30$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{6}$ $30$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{5}$ $15^{2}$ $15^{2}$ $15^{2}$

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
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
19Data not computed