Properties

Label 30.0.675...203.1
Degree $30$
Signature $[0, 15]$
Discriminant $-6.754\times 10^{49}$
Root discriminant $45.81$
Ramified primes $3, 1117$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{30}$ (as 30T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 + 12*x^28 - 45*x^27 + 210*x^26 - 744*x^25 + 2465*x^24 - 6965*x^23 + 19758*x^22 - 48686*x^21 + 121619*x^20 - 306546*x^19 + 810012*x^18 - 2087415*x^17 + 5002572*x^16 - 11311375*x^15 + 24424402*x^14 - 49257771*x^13 + 88022499*x^12 - 134078913*x^11 + 172501038*x^10 - 188691174*x^9 + 176841252*x^8 - 141293241*x^7 + 95362353*x^6 - 55362204*x^5 + 28414476*x^4 - 12173571*x^3 + 3774762*x^2 - 708588*x + 59049)
 
gp: K = bnfinit(x^30 - x^29 + 12*x^28 - 45*x^27 + 210*x^26 - 744*x^25 + 2465*x^24 - 6965*x^23 + 19758*x^22 - 48686*x^21 + 121619*x^20 - 306546*x^19 + 810012*x^18 - 2087415*x^17 + 5002572*x^16 - 11311375*x^15 + 24424402*x^14 - 49257771*x^13 + 88022499*x^12 - 134078913*x^11 + 172501038*x^10 - 188691174*x^9 + 176841252*x^8 - 141293241*x^7 + 95362353*x^6 - 55362204*x^5 + 28414476*x^4 - 12173571*x^3 + 3774762*x^2 - 708588*x + 59049, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59049, -708588, 3774762, -12173571, 28414476, -55362204, 95362353, -141293241, 176841252, -188691174, 172501038, -134078913, 88022499, -49257771, 24424402, -11311375, 5002572, -2087415, 810012, -306546, 121619, -48686, 19758, -6965, 2465, -744, 210, -45, 12, -1, 1]);
 

\( x^{30} - x^{29} + 12 x^{28} - 45 x^{27} + 210 x^{26} - 744 x^{25} + 2465 x^{24} - 6965 x^{23} + 19758 x^{22} - 48686 x^{21} + 121619 x^{20} - 306546 x^{19} + 810012 x^{18} - 2087415 x^{17} + 5002572 x^{16} - 11311375 x^{15} + 24424402 x^{14} - 49257771 x^{13} + 88022499 x^{12} - 134078913 x^{11} + 172501038 x^{10} - 188691174 x^{9} + 176841252 x^{8} - 141293241 x^{7} + 95362353 x^{6} - 55362204 x^{5} + 28414476 x^{4} - 12173571 x^{3} + 3774762 x^{2} - 708588 x + 59049 \)

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-67540340288911750220987142641365428116063232133203\)\(\medspace = -\,3^{15}\cdot 1117^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $45.81$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 1117$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{5}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{6}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{7}$, $\frac{1}{27} a^{14} + \frac{1}{27} a^{13} - \frac{1}{27} a^{12} - \frac{1}{27} a^{11} - \frac{1}{27} a^{8} - \frac{1}{27} a^{7} + \frac{1}{27} a^{6} + \frac{1}{27} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{15} + \frac{1}{27} a^{13} + \frac{1}{27} a^{11} - \frac{1}{27} a^{9} - \frac{1}{27} a^{7} + \frac{8}{27} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{81} a^{16} - \frac{1}{81} a^{15} + \frac{4}{81} a^{13} - \frac{1}{81} a^{12} + \frac{2}{81} a^{10} + \frac{1}{81} a^{9} - \frac{1}{9} a^{8} + \frac{5}{81} a^{7} - \frac{8}{81} a^{6} - \frac{7}{27} a^{4} - \frac{4}{9} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{81} a^{17} - \frac{1}{81} a^{15} + \frac{1}{81} a^{14} + \frac{2}{81} a^{12} - \frac{4}{81} a^{11} + \frac{1}{27} a^{10} + \frac{1}{81} a^{9} - \frac{1}{81} a^{8} - \frac{11}{81} a^{6} + \frac{13}{27} a^{5} - \frac{1}{27} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{81} a^{18} - \frac{1}{27} a^{13} + \frac{4}{81} a^{12} + \frac{1}{27} a^{11} + \frac{1}{27} a^{10} - \frac{1}{9} a^{8} + \frac{1}{27} a^{7} - \frac{5}{81} a^{6} - \frac{1}{27} a^{5} - \frac{10}{27} a^{4} - \frac{1}{3} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{243} a^{19} - \frac{1}{243} a^{17} + \frac{1}{243} a^{16} - \frac{1}{81} a^{15} - \frac{1}{243} a^{14} + \frac{8}{243} a^{13} - \frac{1}{81} a^{12} + \frac{1}{243} a^{11} - \frac{10}{243} a^{10} + \frac{1}{81} a^{9} + \frac{19}{243} a^{8} + \frac{1}{9} a^{7} - \frac{8}{81} a^{6} - \frac{1}{27} a^{5} - \frac{4}{27} a^{4} + \frac{2}{9} a^{3} - \frac{4}{9} a^{2}$, $\frac{1}{243} a^{20} - \frac{1}{243} a^{18} + \frac{1}{243} a^{17} - \frac{4}{243} a^{15} - \frac{1}{243} a^{14} + \frac{7}{243} a^{12} - \frac{1}{243} a^{11} + \frac{1}{27} a^{10} - \frac{5}{243} a^{9} + \frac{1}{27} a^{8} + \frac{13}{81} a^{6} + \frac{4}{27} a^{5} - \frac{1}{27} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2}$, $\frac{1}{729} a^{21} + \frac{1}{729} a^{20} - \frac{1}{729} a^{19} - \frac{1}{243} a^{18} - \frac{2}{729} a^{17} - \frac{1}{729} a^{16} - \frac{5}{729} a^{15} - \frac{4}{729} a^{14} - \frac{26}{729} a^{13} - \frac{5}{243} a^{12} - \frac{16}{729} a^{11} + \frac{19}{729} a^{10} - \frac{23}{729} a^{9} + \frac{4}{243} a^{8} + \frac{11}{81} a^{7} + \frac{11}{81} a^{6} - \frac{8}{27} a^{5} + \frac{2}{27} a^{4} - \frac{5}{27} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{729} a^{22} + \frac{1}{729} a^{20} + \frac{1}{729} a^{19} - \frac{2}{729} a^{18} + \frac{1}{729} a^{17} - \frac{1}{729} a^{16} + \frac{7}{729} a^{15} - \frac{1}{729} a^{14} + \frac{8}{729} a^{13} - \frac{16}{729} a^{12} + \frac{35}{729} a^{11} + \frac{4}{81} a^{10} + \frac{2}{729} a^{9} - \frac{11}{81} a^{8} + \frac{4}{27} a^{7} - \frac{5}{27} a^{5} + \frac{1}{9} a^{4} - \frac{10}{27} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{2187} a^{23} - \frac{1}{2187} a^{22} + \frac{2}{2187} a^{20} + \frac{1}{2187} a^{19} + \frac{1}{729} a^{18} - \frac{1}{243} a^{17} - \frac{2}{729} a^{16} + \frac{1}{729} a^{15} - \frac{29}{2187} a^{14} + \frac{8}{2187} a^{13} + \frac{35}{729} a^{12} - \frac{82}{2187} a^{11} - \frac{119}{2187} a^{10} - \frac{10}{729} a^{9} + \frac{26}{243} a^{8} - \frac{10}{81} a^{7} - \frac{8}{81} a^{6} + \frac{29}{81} a^{5} + \frac{4}{81} a^{4} + \frac{5}{27} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{41553} a^{24} - \frac{1}{41553} a^{23} - \frac{1}{1539} a^{22} - \frac{25}{41553} a^{21} - \frac{71}{41553} a^{20} - \frac{14}{13851} a^{19} - \frac{5}{4617} a^{18} + \frac{61}{13851} a^{17} - \frac{41}{13851} a^{16} + \frac{124}{41553} a^{15} + \frac{584}{41553} a^{14} - \frac{238}{13851} a^{13} - \frac{73}{41553} a^{12} - \frac{2252}{41553} a^{11} + \frac{743}{13851} a^{10} + \frac{55}{1539} a^{9} - \frac{410}{4617} a^{8} + \frac{1}{19} a^{7} + \frac{7}{171} a^{6} + \frac{121}{1539} a^{5} + \frac{34}{513} a^{4} - \frac{20}{171} a^{3} + \frac{26}{171} a^{2} - \frac{1}{57} a + \frac{4}{19}$, $\frac{1}{41553} a^{25} - \frac{1}{4617} a^{23} - \frac{14}{41553} a^{22} + \frac{2}{4617} a^{21} - \frac{25}{13851} a^{20} + \frac{46}{41553} a^{19} - \frac{10}{4617} a^{18} - \frac{37}{13851} a^{17} - \frac{113}{41553} a^{16} + \frac{28}{4617} a^{15} - \frac{227}{13851} a^{14} - \frac{1775}{41553} a^{13} + \frac{109}{4617} a^{12} + \frac{499}{13851} a^{11} - \frac{656}{41553} a^{10} + \frac{5}{171} a^{9} - \frac{5}{1539} a^{8} - \frac{85}{513} a^{7} + \frac{184}{1539} a^{6} - \frac{46}{513} a^{5} - \frac{686}{1539} a^{4} - \frac{70}{171} a^{3} + \frac{4}{171} a^{2} - \frac{9}{19} a + \frac{4}{19}$, $\frac{1}{373977} a^{26} - \frac{1}{373977} a^{24} - \frac{79}{373977} a^{23} - \frac{28}{124659} a^{22} + \frac{124}{373977} a^{21} - \frac{77}{41553} a^{20} + \frac{67}{124659} a^{19} + \frac{125}{41553} a^{18} + \frac{1123}{373977} a^{17} + \frac{20}{41553} a^{16} + \frac{5213}{373977} a^{15} + \frac{845}{373977} a^{14} + \frac{95}{6561} a^{13} - \frac{20690}{373977} a^{12} - \frac{94}{4617} a^{11} + \frac{4810}{124659} a^{10} - \frac{253}{41553} a^{9} + \frac{148}{4617} a^{8} + \frac{8}{1539} a^{7} - \frac{1952}{13851} a^{6} + \frac{44}{1539} a^{5} + \frac{1696}{4617} a^{4} + \frac{509}{1539} a^{3} + \frac{55}{513} a^{2} - \frac{8}{57} a - \frac{7}{19}$, $\frac{1}{373977} a^{27} - \frac{1}{373977} a^{25} + \frac{2}{373977} a^{24} + \frac{2}{124659} a^{23} - \frac{182}{373977} a^{22} - \frac{17}{41553} a^{21} - \frac{197}{124659} a^{20} + \frac{17}{13851} a^{19} + \frac{43}{373977} a^{18} - \frac{214}{41553} a^{17} + \frac{101}{19683} a^{16} + \frac{3707}{373977} a^{15} - \frac{1522}{124659} a^{14} - \frac{10466}{373977} a^{13} - \frac{767}{13851} a^{12} - \frac{134}{124659} a^{11} - \frac{326}{13851} a^{10} - \frac{20}{4617} a^{9} - \frac{59}{513} a^{8} + \frac{163}{13851} a^{7} - \frac{37}{513} a^{6} - \frac{566}{4617} a^{5} + \frac{197}{513} a^{4} + \frac{256}{513} a^{3} + \frac{1}{171} a^{2} + \frac{8}{57} a - \frac{2}{19}$, $\frac{1}{3277160451} a^{28} + \frac{349}{364128939} a^{27} - \frac{1339}{3277160451} a^{26} + \frac{38054}{3277160451} a^{25} + \frac{2083}{1092386817} a^{24} + \frac{326659}{3277160451} a^{23} + \frac{220955}{364128939} a^{22} + \frac{504128}{1092386817} a^{21} - \frac{17485}{19164681} a^{20} + \frac{3371947}{3277160451} a^{19} - \frac{231605}{121376313} a^{18} + \frac{7285577}{3277160451} a^{17} + \frac{4070078}{3277160451} a^{16} + \frac{9836635}{1092386817} a^{15} + \frac{2139280}{172482129} a^{14} + \frac{11753560}{364128939} a^{13} + \frac{25681622}{1092386817} a^{12} + \frac{16747751}{364128939} a^{11} - \frac{983624}{19164681} a^{10} - \frac{3378592}{121376313} a^{9} + \frac{16938748}{121376313} a^{8} - \frac{192614}{13486257} a^{7} - \frac{4735453}{40458771} a^{6} - \frac{4920313}{13486257} a^{5} - \frac{6494456}{13486257} a^{4} + \frac{422411}{4495419} a^{3} - \frac{32230}{65151} a^{2} + \frac{48073}{166497} a + \frac{19340}{55499}$, $\frac{1}{9206095829914799320262012631657534898016816116518072603171} a^{29} + \frac{701662844298639380821432835923510016348132905595}{9206095829914799320262012631657534898016816116518072603171} a^{28} + \frac{627841036211890996904559261616085333070076731723659}{9206095829914799320262012631657534898016816116518072603171} a^{27} + \frac{3242400816987936894289245017403649898100868288705348}{3068698609971599773420670877219178299338938705506024201057} a^{26} + \frac{18646588801344340223996977531976396878603098585283681}{9206095829914799320262012631657534898016816116518072603171} a^{25} + \frac{36615302708783440415052409338260869583894233550420146}{9206095829914799320262012631657534898016816116518072603171} a^{24} + \frac{85639209172559782710872806136481938702188421905322284}{400265036083252144359217940506849343392035483326872721877} a^{23} - \frac{588585575302558707573971103172042073953475468568115334}{1022899536657199924473556959073059433112979568502008067019} a^{22} + \frac{1220023311545997292205750226713971972561689019660330726}{3068698609971599773420670877219178299338938705506024201057} a^{21} + \frac{15687834284024536216831570705261135995452505180963991849}{9206095829914799320262012631657534898016816116518072603171} a^{20} - \frac{1950182121961266066316006726616340810299587916831900665}{9206095829914799320262012631657534898016816116518072603171} a^{19} - \frac{23580447807961243442239924824795981088181664399906827794}{9206095829914799320262012631657534898016816116518072603171} a^{18} + \frac{14772592818894421606982640353798195116352267270009397323}{3068698609971599773420670877219178299338938705506024201057} a^{17} + \frac{15830026367685661255292206669677233823135208633139923606}{9206095829914799320262012631657534898016816116518072603171} a^{16} - \frac{42611591145227208084400416595157291354328190031496962441}{9206095829914799320262012631657534898016816116518072603171} a^{15} - \frac{15866161821928172331427706671852750167453300583745343484}{9206095829914799320262012631657534898016816116518072603171} a^{14} - \frac{50953715193343182112097193029064835161544257450243369017}{1022899536657199924473556959073059433112979568502008067019} a^{13} + \frac{7258179032776301830342125813897325515154202049488372941}{161510453156399988074772151432588331544154668710843379003} a^{12} + \frac{258283760422604875088705415292155130920892048728646660}{12628389341446912647821690852753820161888636648172939099} a^{11} + \frac{51675541383211414842391272838758374906967565010586332339}{1022899536657199924473556959073059433112979568502008067019} a^{10} + \frac{14202075901144722662429266788946248936847174278390969248}{340966512219066641491185653024353144370993189500669355673} a^{9} - \frac{20187881767983756923850698562791599951856082348046675783}{340966512219066641491185653024353144370993189500669355673} a^{8} + \frac{1266441051546162637390641484228807146646734522384632359}{37885168024340737943465072558261460485665909944518817297} a^{7} - \frac{13141615789694571350841945607908148185449908626222862007}{113655504073022213830395217674784381456997729833556451891} a^{6} + \frac{469775585685755117629056399385116698394874601379892636}{4209463113815637549273896950917940053962878882724313033} a^{5} - \frac{7738711813544814805569990747460011420688344845852407942}{37885168024340737943465072558261460485665909944518817297} a^{4} + \frac{1231348409886560915858733937893418121408607036288039073}{12628389341446912647821690852753820161888636648172939099} a^{3} + \frac{1249390381399919788367031903006655660889956446661045640}{4209463113815637549273896950917940053962878882724313033} a^{2} - \frac{144969404927624234261313161617902332092013960500759652}{467718123757293061030432994546437783773653209191590337} a + \frac{42969898752279757971925717674794138466787849498409096}{155906041252431020343477664848812594591217736397196779}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{17240303365876863765732656419376340682469}{59058307037381865563052279638600485479572673} a^{29} + \frac{10166566551583970140711423627320830797232}{59058307037381865563052279638600485479572673} a^{28} - \frac{67678054620361900780146076046996910202402}{19686102345793955187684093212866828493190891} a^{27} + \frac{230829970243523141033020719991861178114737}{19686102345793955187684093212866828493190891} a^{26} - \frac{1113395891465648914640434494394938747891794}{19686102345793955187684093212866828493190891} a^{25} + \frac{3822240881692960369901247648111492149173117}{19686102345793955187684093212866828493190891} a^{24} - \frac{37848747529731382545608466649026300153743839}{59058307037381865563052279638600485479572673} a^{23} + \frac{104728577812217625409984431483005409814057531}{59058307037381865563052279638600485479572673} a^{22} - \frac{33140300279142901862255430506119199291802538}{6562034115264651729228031070955609497730297} a^{21} + \frac{718577946267968546054639497068341451751309786}{59058307037381865563052279638600485479572673} a^{20} - \frac{1806539382225062528177134129976064227160314974}{59058307037381865563052279638600485479572673} a^{19} + \frac{1518124354046899318167390245430748440087923000}{19686102345793955187684093212866828493190891} a^{18} - \frac{4041228218252847128907407260893834677055345163}{19686102345793955187684093212866828493190891} a^{17} + \frac{16683862372054993358864746794673323917019413}{31700647899829235406898700825872509650871} a^{16} - \frac{24559530661140143253979389375113794120457830973}{19686102345793955187684093212866828493190891} a^{15} + \frac{165250949628186772549645709958918208645653101843}{59058307037381865563052279638600485479572673} a^{14} - \frac{354379390190139123097533623815978113935669118890}{59058307037381865563052279638600485479572673} a^{13} + \frac{78473111054066411257168767350698673618290533971}{6562034115264651729228031070955609497730297} a^{12} - \frac{410981901890143514005765414123533052449595104767}{19686102345793955187684093212866828493190891} a^{11} + \frac{67253612884774521743965756138194203741805595210}{2187344705088217243076010356985203165910099} a^{10} - \frac{27729354552007667643772408710133672929492078329}{729114901696072414358670118995067721970033} a^{9} + \frac{87248112007248015454744277625518328697967416088}{2187344705088217243076010356985203165910099} a^{8} - \frac{78176582382846781161047682008722997620533228728}{2187344705088217243076010356985203165910099} a^{7} + \frac{15051782725862379476775773291366471061664238}{556151717540863779068398260103026485103} a^{6} - \frac{12499233710878423811675624348901580021467635657}{729114901696072414358670118995067721970033} a^{5} + \frac{762835590123433623570483732659219489381153540}{81012766855119157150963346555007524663337} a^{4} - \frac{41406321804769367425051723860403850514482815}{9001418539457684127884816283889724962593} a^{3} + \frac{47338291813799531068085097058835292988823156}{27004255618373052383654448851669174887779} a^{2} - \frac{3793307478431774579124009715514562797941385}{9001418539457684127884816283889724962593} a + \frac{138764118402052272380358033203706218271370}{3000472846485894709294938761296574987531} \) (order $6$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 3232250515402658.0 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 3232250515402658.0 \cdot 1}{6\sqrt{67540340288911750220987142641365428116063232133203}}\approx 61.5559681346093$ (assuming GRH)

Galois group

$D_{30}$ (as 30T14):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 60
The 18 conjugacy class representatives for $D_{30}$
Character table for $D_{30}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.3351.1, 5.1.11229201.1, 6.0.33687603.2, 10.0.378284865295203.1, 15.1.4744833691813716605070951.1

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

Sibling fields

Degree 30 sibling: 30.2.25147520034238141665614212776801727735214210097595917.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $30$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{6}$ $30$ $15^{2}$ $30$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{5}$ $15^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{15}$ $30$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
1117Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.1117.2t1.a.a$1$ $ 1117 $ $x^{2} - x - 279$ $C_2$ (as 2T1) $1$ $1$
1.3351.2t1.a.a$1$ $ 3 \cdot 1117 $ $x^{2} - x + 838$ $C_2$ (as 2T1) $1$ $-1$
* 1.3.2t1.a.a$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
* 2.3351.6t3.b.a$2$ $ 3 \cdot 1117 $ $x^{6} - 35 x^{3} + 27$ $D_{6}$ (as 6T3) $1$ $0$
* 2.3351.3t2.a.a$2$ $ 3 \cdot 1117 $ $x^{3} - x^{2} + 8 x - 35$ $S_3$ (as 3T2) $1$ $0$
* 2.3351.5t2.a.b$2$ $ 3 \cdot 1117 $ $x^{5} - x^{4} - 2 x^{3} + 9 x^{2} + 21 x + 9$ $D_{5}$ (as 5T2) $1$ $0$
* 2.3351.5t2.a.a$2$ $ 3 \cdot 1117 $ $x^{5} - x^{4} - 2 x^{3} + 9 x^{2} + 21 x + 9$ $D_{5}$ (as 5T2) $1$ $0$
* 2.3351.10t3.a.a$2$ $ 3 \cdot 1117 $ $x^{10} - 2 x^{9} + 11 x^{8} + 79 x^{7} + 140 x^{6} + 247 x^{5} + 481 x^{4} - 945 x^{3} + 618 x^{2} + 612 x + 9$ $D_{10}$ (as 10T3) $1$ $0$
* 2.3351.10t3.a.b$2$ $ 3 \cdot 1117 $ $x^{10} - 2 x^{9} + 11 x^{8} + 79 x^{7} + 140 x^{6} + 247 x^{5} + 481 x^{4} - 945 x^{3} + 618 x^{2} + 612 x + 9$ $D_{10}$ (as 10T3) $1$ $0$
* 2.3351.15t2.a.b$2$ $ 3 \cdot 1117 $ $x^{15} - 3 x^{14} - 3 x^{13} + 5 x^{12} + 99 x^{11} - 453 x^{10} + 1246 x^{9} - 2652 x^{8} + 5058 x^{7} - 7317 x^{6} + 8748 x^{5} - 9882 x^{4} + 8451 x^{3} - 4536 x^{2} + 1944 x - 729$ $D_{15}$ (as 15T2) $1$ $0$
* 2.3351.15t2.a.a$2$ $ 3 \cdot 1117 $ $x^{15} - 3 x^{14} - 3 x^{13} + 5 x^{12} + 99 x^{11} - 453 x^{10} + 1246 x^{9} - 2652 x^{8} + 5058 x^{7} - 7317 x^{6} + 8748 x^{5} - 9882 x^{4} + 8451 x^{3} - 4536 x^{2} + 1944 x - 729$ $D_{15}$ (as 15T2) $1$ $0$
* 2.3351.15t2.a.d$2$ $ 3 \cdot 1117 $ $x^{15} - 3 x^{14} - 3 x^{13} + 5 x^{12} + 99 x^{11} - 453 x^{10} + 1246 x^{9} - 2652 x^{8} + 5058 x^{7} - 7317 x^{6} + 8748 x^{5} - 9882 x^{4} + 8451 x^{3} - 4536 x^{2} + 1944 x - 729$ $D_{15}$ (as 15T2) $1$ $0$
* 2.3351.15t2.a.c$2$ $ 3 \cdot 1117 $ $x^{15} - 3 x^{14} - 3 x^{13} + 5 x^{12} + 99 x^{11} - 453 x^{10} + 1246 x^{9} - 2652 x^{8} + 5058 x^{7} - 7317 x^{6} + 8748 x^{5} - 9882 x^{4} + 8451 x^{3} - 4536 x^{2} + 1944 x - 729$ $D_{15}$ (as 15T2) $1$ $0$
* 2.3351.30t14.b.b$2$ $ 3 \cdot 1117 $ $x^{30} - x^{29} + 12 x^{28} - 45 x^{27} + 210 x^{26} - 744 x^{25} + 2465 x^{24} - 6965 x^{23} + 19758 x^{22} - 48686 x^{21} + 121619 x^{20} - 306546 x^{19} + 810012 x^{18} - 2087415 x^{17} + 5002572 x^{16} - 11311375 x^{15} + 24424402 x^{14} - 49257771 x^{13} + 88022499 x^{12} - 134078913 x^{11} + 172501038 x^{10} - 188691174 x^{9} + 176841252 x^{8} - 141293241 x^{7} + 95362353 x^{6} - 55362204 x^{5} + 28414476 x^{4} - 12173571 x^{3} + 3774762 x^{2} - 708588 x + 59049$ $D_{30}$ (as 30T14) $1$ $0$
* 2.3351.30t14.b.c$2$ $ 3 \cdot 1117 $ $x^{30} - x^{29} + 12 x^{28} - 45 x^{27} + 210 x^{26} - 744 x^{25} + 2465 x^{24} - 6965 x^{23} + 19758 x^{22} - 48686 x^{21} + 121619 x^{20} - 306546 x^{19} + 810012 x^{18} - 2087415 x^{17} + 5002572 x^{16} - 11311375 x^{15} + 24424402 x^{14} - 49257771 x^{13} + 88022499 x^{12} - 134078913 x^{11} + 172501038 x^{10} - 188691174 x^{9} + 176841252 x^{8} - 141293241 x^{7} + 95362353 x^{6} - 55362204 x^{5} + 28414476 x^{4} - 12173571 x^{3} + 3774762 x^{2} - 708588 x + 59049$ $D_{30}$ (as 30T14) $1$ $0$
* 2.3351.30t14.b.d$2$ $ 3 \cdot 1117 $ $x^{30} - x^{29} + 12 x^{28} - 45 x^{27} + 210 x^{26} - 744 x^{25} + 2465 x^{24} - 6965 x^{23} + 19758 x^{22} - 48686 x^{21} + 121619 x^{20} - 306546 x^{19} + 810012 x^{18} - 2087415 x^{17} + 5002572 x^{16} - 11311375 x^{15} + 24424402 x^{14} - 49257771 x^{13} + 88022499 x^{12} - 134078913 x^{11} + 172501038 x^{10} - 188691174 x^{9} + 176841252 x^{8} - 141293241 x^{7} + 95362353 x^{6} - 55362204 x^{5} + 28414476 x^{4} - 12173571 x^{3} + 3774762 x^{2} - 708588 x + 59049$ $D_{30}$ (as 30T14) $1$ $0$
* 2.3351.30t14.b.a$2$ $ 3 \cdot 1117 $ $x^{30} - x^{29} + 12 x^{28} - 45 x^{27} + 210 x^{26} - 744 x^{25} + 2465 x^{24} - 6965 x^{23} + 19758 x^{22} - 48686 x^{21} + 121619 x^{20} - 306546 x^{19} + 810012 x^{18} - 2087415 x^{17} + 5002572 x^{16} - 11311375 x^{15} + 24424402 x^{14} - 49257771 x^{13} + 88022499 x^{12} - 134078913 x^{11} + 172501038 x^{10} - 188691174 x^{9} + 176841252 x^{8} - 141293241 x^{7} + 95362353 x^{6} - 55362204 x^{5} + 28414476 x^{4} - 12173571 x^{3} + 3774762 x^{2} - 708588 x + 59049$ $D_{30}$ (as 30T14) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.