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Properties

Label 40.0.11302165783...0000.9
Degree $40$
Signature $[0, 20]$
Discriminant $2^{40}\cdot 3^{20}\cdot 5^{20}\cdot 11^{36}$
Root discriminant $67.04$
Ramified primes $2, 3, 5, 11$
Class number Not computed
Class group Not computed
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1568239201, 0, 1563447480, 0, 43140130, 0, -1146406702, 0, -814899493, 0, -101113408, 0, 89053119, 0, 60919057, 0, 44628595, 0, 33266232, 0, 20212367, 0, 10028666, 0, 4119588, 0, 1406098, 0, 398388, 0, 92906, 0, 17588, 0, 2628, 0, 297, 0, 23, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 23*x^38 + 297*x^36 + 2628*x^34 + 17588*x^32 + 92906*x^30 + 398388*x^28 + 1406098*x^26 + 4119588*x^24 + 10028666*x^22 + 20212367*x^20 + 33266232*x^18 + 44628595*x^16 + 60919057*x^14 + 89053119*x^12 - 101113408*x^10 - 814899493*x^8 - 1146406702*x^6 + 43140130*x^4 + 1563447480*x^2 + 1568239201)
 
gp: K = bnfinit(x^40 + 23*x^38 + 297*x^36 + 2628*x^34 + 17588*x^32 + 92906*x^30 + 398388*x^28 + 1406098*x^26 + 4119588*x^24 + 10028666*x^22 + 20212367*x^20 + 33266232*x^18 + 44628595*x^16 + 60919057*x^14 + 89053119*x^12 - 101113408*x^10 - 814899493*x^8 - 1146406702*x^6 + 43140130*x^4 + 1563447480*x^2 + 1568239201, 1)
 

Normalized defining polynomial

\( x^{40} + 23 x^{38} + 297 x^{36} + 2628 x^{34} + 17588 x^{32} + 92906 x^{30} + 398388 x^{28} + 1406098 x^{26} + 4119588 x^{24} + 10028666 x^{22} + 20212367 x^{20} + 33266232 x^{18} + 44628595 x^{16} + 60919057 x^{14} + 89053119 x^{12} - 101113408 x^{10} - 814899493 x^{8} - 1146406702 x^{6} + 43140130 x^{4} + 1563447480 x^{2} + 1568239201 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11302165783522556415463223790320401501047994110286233600000000000000000000=2^{40}\cdot 3^{20}\cdot 5^{20}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.04$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(660=2^{2}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{660}(1,·)$, $\chi_{660}(259,·)$, $\chi_{660}(641,·)$, $\chi_{660}(521,·)$, $\chi_{660}(139,·)$, $\chi_{660}(401,·)$, $\chi_{660}(19,·)$, $\chi_{660}(149,·)$, $\chi_{660}(409,·)$, $\chi_{660}(29,·)$, $\chi_{660}(31,·)$, $\chi_{660}(421,·)$, $\chi_{660}(551,·)$, $\chi_{660}(299,·)$, $\chi_{660}(301,·)$, $\chi_{660}(221,·)$, $\chi_{660}(181,·)$, $\chi_{660}(311,·)$, $\chi_{660}(569,·)$, $\chi_{660}(191,·)$, $\chi_{660}(331,·)$, $\chi_{660}(581,·)$, $\chi_{660}(71,·)$, $\chi_{660}(329,·)$, $\chi_{660}(439,·)$, $\chi_{660}(589,·)$, $\chi_{660}(79,·)$, $\chi_{660}(469,·)$, $\chi_{660}(91,·)$, $\chi_{660}(349,·)$, $\chi_{660}(479,·)$, $\chi_{660}(359,·)$, $\chi_{660}(361,·)$, $\chi_{660}(109,·)$, $\chi_{660}(239,·)$, $\chi_{660}(659,·)$, $\chi_{660}(629,·)$, $\chi_{660}(631,·)$, $\chi_{660}(251,·)$, $\chi_{660}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{199} a^{21} + \frac{11}{199} a^{19} + \frac{77}{199} a^{17} - \frac{68}{199} a^{15} + \frac{39}{199} a^{13} + \frac{5}{199} a^{9} + \frac{83}{199} a^{7} - \frac{11}{199} a^{5} + \frac{8}{199} a^{3} - \frac{78}{199} a$, $\frac{1}{199} a^{22} + \frac{11}{199} a^{20} + \frac{77}{199} a^{18} - \frac{68}{199} a^{16} + \frac{39}{199} a^{14} + \frac{5}{199} a^{10} + \frac{83}{199} a^{8} - \frac{11}{199} a^{6} + \frac{8}{199} a^{4} - \frac{78}{199} a^{2}$, $\frac{1}{199} a^{23} - \frac{44}{199} a^{19} + \frac{80}{199} a^{17} - \frac{9}{199} a^{15} - \frac{31}{199} a^{13} + \frac{5}{199} a^{11} + \frac{28}{199} a^{9} + \frac{71}{199} a^{7} - \frac{70}{199} a^{5} + \frac{33}{199} a^{3} + \frac{62}{199} a$, $\frac{1}{17711} a^{24} + \frac{13}{17711} a^{22} + \frac{2686}{17711} a^{20} + \frac{6056}{17711} a^{18} + \frac{4878}{17711} a^{16} + \frac{277}{17711} a^{14} - \frac{2980}{17711} a^{12} - \frac{902}{17711} a^{10} - \frac{4422}{17711} a^{8} + \frac{981}{17711} a^{6} + \frac{2923}{17711} a^{4} + \frac{7008}{17711} a^{2} + \frac{5}{89}$, $\frac{1}{17711} a^{25} + \frac{13}{17711} a^{23} + \frac{16}{17711} a^{21} - \frac{5603}{17711} a^{19} - \frac{5891}{17711} a^{17} + \frac{4727}{17711} a^{15} - \frac{844}{17711} a^{13} - \frac{902}{17711} a^{11} - \frac{61}{17711} a^{9} - \frac{8097}{17711} a^{7} - \frac{3129}{17711} a^{5} + \frac{3359}{17711} a^{3} - \frac{3277}{17711} a$, $\frac{1}{17711} a^{26} + \frac{25}{17711} a^{22} - \frac{3141}{17711} a^{20} - \frac{69}{17711} a^{18} + \frac{53}{17711} a^{16} + \frac{2497}{17711} a^{14} + \frac{2416}{17711} a^{12} - \frac{5156}{17711} a^{10} - \frac{6681}{17711} a^{8} - \frac{129}{17711} a^{6} + \frac{2206}{17711} a^{4} - \frac{1999}{17711} a^{2} + \frac{24}{89}$, $\frac{1}{17711} a^{27} + \frac{25}{17711} a^{23} - \frac{26}{17711} a^{21} - \frac{1226}{17711} a^{19} - \frac{8046}{17711} a^{17} + \frac{3209}{17711} a^{15} - \frac{76}{17711} a^{13} - \frac{5156}{17711} a^{11} - \frac{8817}{17711} a^{9} - \frac{7249}{17711} a^{7} + \frac{3363}{17711} a^{5} + \frac{5210}{17711} a^{3} - \frac{7951}{17711} a$, $\frac{1}{17711} a^{28} + \frac{5}{17711} a^{22} + \frac{6384}{17711} a^{20} - \frac{8057}{17711} a^{18} - \frac{1261}{17711} a^{16} + \frac{6883}{17711} a^{14} - \frac{1500}{17711} a^{12} - \frac{2198}{17711} a^{10} - \frac{8839}{17711} a^{8} - \frac{7367}{17711} a^{6} + \frac{5827}{17711} a^{4} + \frac{1613}{17711} a^{2} - \frac{36}{89}$, $\frac{1}{17711} a^{29} + \frac{5}{17711} a^{23} - \frac{24}{17711} a^{21} - \frac{7701}{17711} a^{19} + \frac{1231}{17711} a^{17} - \frac{148}{17711} a^{15} - \frac{3458}{17711} a^{13} - \frac{2198}{17711} a^{11} - \frac{5457}{17711} a^{9} - \frac{7901}{17711} a^{7} + \frac{5471}{17711} a^{5} + \frac{3482}{17711} a^{3} - \frac{3248}{17711} a$, $\frac{1}{761573} a^{30} - \frac{21}{761573} a^{28} + \frac{1}{761573} a^{26} + \frac{18}{761573} a^{24} + \frac{16}{17711} a^{22} - \frac{156268}{761573} a^{20} + \frac{226214}{761573} a^{18} - \frac{200518}{761573} a^{16} - \frac{153028}{761573} a^{14} + \frac{364909}{761573} a^{12} - \frac{238731}{761573} a^{10} - \frac{11850}{761573} a^{8} + \frac{272215}{761573} a^{6} - \frac{179962}{761573} a^{4} + \frac{215922}{761573} a^{2} - \frac{1380}{3827}$, $\frac{1}{761573} a^{31} - \frac{21}{761573} a^{29} + \frac{1}{761573} a^{27} + \frac{18}{761573} a^{25} + \frac{16}{17711} a^{23} + \frac{639}{761573} a^{21} - \frac{332528}{761573} a^{19} - \frac{303847}{761573} a^{17} - \frac{160682}{761573} a^{15} - \frac{369875}{761573} a^{13} - \frac{238731}{761573} a^{11} + \frac{11112}{761573} a^{9} + \frac{348755}{761573} a^{7} + \frac{378780}{761573} a^{5} - \frac{51968}{761573} a^{3} - \frac{328198}{761573} a$, $\frac{1}{761573} a^{32} - \frac{10}{761573} a^{28} - \frac{4}{761573} a^{26} - \frac{9}{761573} a^{24} - \frac{1640}{761573} a^{22} + \frac{144345}{761573} a^{20} + \frac{272809}{761573} a^{18} + \frac{766}{761573} a^{16} + \frac{344974}{761573} a^{14} - \frac{21479}{761573} a^{12} - \frac{205718}{761573} a^{10} + \frac{260854}{761573} a^{8} - \frac{364165}{761573} a^{6} - \frac{23821}{761573} a^{4} + \frac{35336}{761573} a^{2} - \frac{1116}{3827}$, $\frac{1}{761573} a^{33} - \frac{10}{761573} a^{29} - \frac{4}{761573} a^{27} - \frac{9}{761573} a^{25} - \frac{1640}{761573} a^{23} - \frac{1081}{761573} a^{21} + \frac{196269}{761573} a^{19} + \frac{226559}{761573} a^{17} + \frac{333493}{761573} a^{15} - \frac{362082}{761573} a^{13} - \frac{205718}{761573} a^{11} + \frac{295297}{761573} a^{9} - \frac{249355}{761573} a^{7} + \frac{52719}{761573} a^{5} - \frac{366499}{761573} a^{3} - \frac{302451}{761573} a$, $\frac{1}{761573} a^{34} + \frac{1}{761573} a^{28} + \frac{1}{761573} a^{26} + \frac{2}{761573} a^{24} - \frac{909}{761573} a^{22} - \frac{3941}{17711} a^{20} - \frac{68167}{761573} a^{18} + \frac{156329}{761573} a^{16} - \frac{290741}{761573} a^{14} + \frac{287258}{761573} a^{12} - \frac{209387}{761573} a^{10} - \frac{294669}{761573} a^{8} - \frac{126427}{761573} a^{6} + \frac{99508}{761573} a^{4} + \frac{68915}{761573} a^{2} + \frac{1078}{3827}$, $\frac{1}{761573} a^{35} + \frac{1}{761573} a^{29} + \frac{1}{761573} a^{27} + \frac{2}{761573} a^{25} - \frac{909}{761573} a^{23} - \frac{25}{17711} a^{21} + \frac{260955}{761573} a^{19} + \frac{175464}{761573} a^{17} - \frac{317530}{761573} a^{15} + \frac{233}{761573} a^{13} - \frac{209387}{761573} a^{11} - \frac{214302}{761573} a^{9} + \frac{141463}{761573} a^{7} - \frac{229614}{761573} a^{5} - \frac{107127}{761573} a^{3} + \frac{26999}{761573} a$, $\frac{1}{761573} a^{36} - \frac{21}{761573} a^{28} + \frac{1}{761573} a^{26} + \frac{19}{761573} a^{24} - \frac{27}{17711} a^{22} + \frac{272657}{761573} a^{20} - \frac{2139}{8557} a^{18} + \frac{1496}{761573} a^{16} - \frac{328425}{761573} a^{14} - \frac{282584}{761573} a^{12} - \frac{30181}{761573} a^{10} - \frac{33307}{761573} a^{8} + \frac{107696}{761573} a^{6} + \frac{210865}{761573} a^{4} - \frac{348926}{761573} a^{2} + \frac{4}{3827}$, $\frac{1}{761573} a^{37} - \frac{21}{761573} a^{29} + \frac{1}{761573} a^{27} + \frac{19}{761573} a^{25} - \frac{27}{17711} a^{23} + \frac{940}{761573} a^{21} - \frac{1494}{8557} a^{19} - \frac{358242}{761573} a^{17} - \frac{129421}{761573} a^{15} - \frac{217525}{761573} a^{13} - \frac{30181}{761573} a^{11} + \frac{131254}{761573} a^{9} - \frac{359198}{761573} a^{7} + \frac{153460}{761573} a^{5} - \frac{237943}{761573} a^{3} - \frac{129322}{761573} a$, $\frac{1}{579874362828688973077578404795795486583718302294159513307} a^{38} - \frac{329233038513584965931955440966918808963246994711036}{579874362828688973077578404795795486583718302294159513307} a^{36} + \frac{165496344442966150370799006052850978368865753663672}{579874362828688973077578404795795486583718302294159513307} a^{34} - \frac{282475462724700954155930174042154865749377962849350}{579874362828688973077578404795795486583718302294159513307} a^{32} - \frac{150976496226431481127872045515283371753933266079876}{579874362828688973077578404795795486583718302294159513307} a^{30} + \frac{11875028897717226947534729459368597403045632788509026}{579874362828688973077578404795795486583718302294159513307} a^{28} - \frac{14573379497380385005024307074521772247215203218196966}{579874362828688973077578404795795486583718302294159513307} a^{26} - \frac{14432109695145939401178260507066891175466252865821771}{579874362828688973077578404795795486583718302294159513307} a^{24} - \frac{29274505187056699335963660429639740141929903383677501}{579874362828688973077578404795795486583718302294159513307} a^{22} + \frac{179863314407490325981130726374119700715466065106827396348}{579874362828688973077578404795795486583718302294159513307} a^{20} - \frac{183938309081489572646961338210595784685355028920191547679}{579874362828688973077578404795795486583718302294159513307} a^{18} + \frac{275281457591285191704443286038319869280263752957815687453}{579874362828688973077578404795795486583718302294159513307} a^{16} + \frac{226196031227081452308003221671495628642481905564882506944}{579874362828688973077578404795795486583718302294159513307} a^{14} - \frac{235453672275359698355249829577640537803634950285923905404}{579874362828688973077578404795795486583718302294159513307} a^{12} + \frac{222233579375812131905060043371564075662331981957391975734}{579874362828688973077578404795795486583718302294159513307} a^{10} + \frac{199070375058689867564638460159698852571573625852922936915}{579874362828688973077578404795795486583718302294159513307} a^{8} - \frac{13306629287453231797675958690053853136885500412735619202}{579874362828688973077578404795795486583718302294159513307} a^{6} + \frac{18832729862781449727070228405884778410544052312325059649}{579874362828688973077578404795795486583718302294159513307} a^{4} - \frac{18046406926718794632073656900263023058813204402251727081}{579874362828688973077578404795795486583718302294159513307} a^{2} + \frac{3610356328123702530158487429272060321754070566729793}{14642922219860331129960819292335938147615421385676107}$, $\frac{1}{115394998202909105642438102554363301830159942156537743148093} a^{39} + \frac{49924265899465403552726367157815472523877344716673858}{115394998202909105642438102554363301830159942156537743148093} a^{37} + \frac{64124494992779860628663209586321349038347800659062628}{115394998202909105642438102554363301830159942156537743148093} a^{35} - \frac{47490307798401932592895566554716522481448115631120008}{115394998202909105642438102554363301830159942156537743148093} a^{33} - \frac{26039142615791364960436704899433484967459692632550882}{115394998202909105642438102554363301830159942156537743148093} a^{31} + \frac{1018467840958447875760660524544070435918428394037764024}{115394998202909105642438102554363301830159942156537743148093} a^{29} + \frac{851918768857469447331841921977210975279642272047803176}{115394998202909105642438102554363301830159942156537743148093} a^{27} - \frac{315191686672444431293148524545234233243224339623352576}{115394998202909105642438102554363301830159942156537743148093} a^{25} + \frac{230857380698755104292759780833457949214877026860531103746}{115394998202909105642438102554363301830159942156537743148093} a^{23} + \frac{170341799192044458535284933584759491569751343904541516053}{115394998202909105642438102554363301830159942156537743148093} a^{21} - \frac{24621436674803783590387684393669746089786005034458649812581}{115394998202909105642438102554363301830159942156537743148093} a^{19} - \frac{39503958361243101444810363059978419392661150545301224269145}{115394998202909105642438102554363301830159942156537743148093} a^{17} + \frac{23988510466301556468049412465168487006967034999310985914273}{115394998202909105642438102554363301830159942156537743148093} a^{15} + \frac{27288348102389902390474474514854639790884834285937022053328}{115394998202909105642438102554363301830159942156537743148093} a^{13} - \frac{41365071547831566504448014233725885155001848967240989632177}{115394998202909105642438102554363301830159942156537743148093} a^{11} - \frac{46043647320597908173870264733362825572609606905918142066610}{115394998202909105642438102554363301830159942156537743148093} a^{9} - \frac{17070488578063336820177940095063251561414698378913468816979}{115394998202909105642438102554363301830159942156537743148093} a^{7} + \frac{46703648729676394463701890197947206548289026120492701661108}{115394998202909105642438102554363301830159942156537743148093} a^{5} - \frac{43560085486240833986011220077940741266580302498767205692239}{115394998202909105642438102554363301830159942156537743148093} a^{3} + \frac{69302630065458958649912160392458818110206950046916122}{2913941521752205894862203039174851691375468855749545293} a$

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:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1590157875862031017547563145397333409561008184753}{1296573013515832647667843848925430357642246541084693743237} a^{39} + \frac{31878426119377273066402501245807561796778114908133}{1296573013515832647667843848925430357642246541084693743237} a^{37} + \frac{374439450319334025720228803563089241475900794060683}{1296573013515832647667843848925430357642246541084693743237} a^{35} + \frac{3001006376380458462578066901380267109649159168377106}{1296573013515832647667843848925430357642246541084693743237} a^{33} + \frac{18279251711100130688749005827567231068406443003500910}{1296573013515832647667843848925430357642246541084693743237} a^{31} + \frac{87353194031454943749210676557360391845329664889627249}{1296573013515832647667843848925430357642246541084693743237} a^{29} + \frac{338145706585217658940376637722210624584903430881355513}{1296573013515832647667843848925430357642246541084693743237} a^{27} + \frac{1068659207205242925052141680286937928945399895602586477}{1296573013515832647667843848925430357642246541084693743237} a^{25} + \frac{2792457628391738638985817125945483241436502297433193647}{1296573013515832647667843848925430357642246541084693743237} a^{23} + \frac{6018530457546996330111582239519680621143923390274782288}{1296573013515832647667843848925430357642246541084693743237} a^{21} + \frac{10793469221090474551909546509477769905523108716193981621}{1296573013515832647667843848925430357642246541084693743237} a^{19} + \frac{16020883084121789246588763073191330671872496261876770580}{1296573013515832647667843848925430357642246541084693743237} a^{17} + \frac{22394288678535123955232880792921973869109295513774486510}{1296573013515832647667843848925430357642246541084693743237} a^{15} + \frac{49124828703616508567226463205948899465139705931508627038}{1296573013515832647667843848925430357642246541084693743237} a^{13} + \frac{62483075973796180168809932491127539906878949630212830859}{1296573013515832647667843848925430357642246541084693743237} a^{11} - \frac{224648952742610269058561386661054660521300095900531209760}{1296573013515832647667843848925430357642246541084693743237} a^{9} - \frac{310725495395793283399427567449049196766284465291006721746}{1296573013515832647667843848925430357642246541084693743237} a^{7} + \frac{447632461604095755214794508040983728026740912457702009642}{1296573013515832647667843848925430357642246541084693743237} a^{5} + \frac{313607839972982061622518701719182426746478016405554742446}{1296573013515832647667843848925430357642246541084693743237} a^{3} + \frac{9762959741367949196257401177335463258163940714758074}{32740915974743886459125876844661254959274930963478037} a \) (order $12$)
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

$C_2^2\times C_{10}$ (as 40T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-165}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(i, \sqrt{165})\), \(\Q(i, \sqrt{55})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-3}, \sqrt{55})\), \(\Q(\sqrt{3}, \sqrt{-55})\), \(\Q(\sqrt{3}, \sqrt{55})\), \(\Q(\sqrt{-3}, \sqrt{-55})\), \(\Q(\zeta_{11})^+\), 8.0.189747360000.10, 10.0.219503494144.1, 10.0.1833540124521600000.1, 10.10.1790566527853125.1, 10.10.7545432611200000.1, 10.0.7368586534375.1, 10.0.52089208083.1, 10.10.53339349076992.1, 20.0.3361869388230684433628866560000000000.3, 20.0.56933553290160450365440000000000.2, 20.0.2845086159957207322343768064.1, 20.0.3361869388230684433628866560000000000.9, 20.0.3361869388230684433628866560000000000.4, 20.20.3361869388230684433628866560000000000.1, 20.0.3206128490667995866421572265625.2

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 R R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
11Data not computed