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Properties

Label 40.0.68563211152...5625.1
Degree $40$
Signature $[0, 20]$
Discriminant $3^{20}\cdot 5^{30}\cdot 11^{32}$
Root discriminant $39.44$
Ramified primes $3, 5, 11$
Class number $55$ (GRH)
Class group $[55]$ (GRH)
Galois group $C_2\times C_{20}$ (as 40T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -3, 46, -125, 406, 393, -7029, 19004, 19768, -17836, -33345, -3560, -25677, 12498, 82301, 39515, -96822, 2424, 43346, -17018, 16639, -3176, -19250, 11925, 6247, -3261, -544, 119, -676, 182, 377, -80, -99, 43, -1, -1, 7, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 - 4*x^38 + 7*x^37 - x^36 - x^35 + 43*x^34 - 99*x^33 - 80*x^32 + 377*x^31 + 182*x^30 - 676*x^29 + 119*x^28 - 544*x^27 - 3261*x^26 + 6247*x^25 + 11925*x^24 - 19250*x^23 - 3176*x^22 + 16639*x^21 - 17018*x^20 + 43346*x^19 + 2424*x^18 - 96822*x^17 + 39515*x^16 + 82301*x^15 + 12498*x^14 - 25677*x^13 - 3560*x^12 - 33345*x^11 - 17836*x^10 + 19768*x^9 + 19004*x^8 - 7029*x^7 + 393*x^6 + 406*x^5 - 125*x^4 + 46*x^3 - 3*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^40 - x^39 - 4*x^38 + 7*x^37 - x^36 - x^35 + 43*x^34 - 99*x^33 - 80*x^32 + 377*x^31 + 182*x^30 - 676*x^29 + 119*x^28 - 544*x^27 - 3261*x^26 + 6247*x^25 + 11925*x^24 - 19250*x^23 - 3176*x^22 + 16639*x^21 - 17018*x^20 + 43346*x^19 + 2424*x^18 - 96822*x^17 + 39515*x^16 + 82301*x^15 + 12498*x^14 - 25677*x^13 - 3560*x^12 - 33345*x^11 - 17836*x^10 + 19768*x^9 + 19004*x^8 - 7029*x^7 + 393*x^6 + 406*x^5 - 125*x^4 + 46*x^3 - 3*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} - 4 x^{38} + 7 x^{37} - x^{36} - x^{35} + 43 x^{34} - 99 x^{33} - 80 x^{32} + 377 x^{31} + 182 x^{30} - 676 x^{29} + 119 x^{28} - 544 x^{27} - 3261 x^{26} + 6247 x^{25} + 11925 x^{24} - 19250 x^{23} - 3176 x^{22} + 16639 x^{21} - 17018 x^{20} + 43346 x^{19} + 2424 x^{18} - 96822 x^{17} + 39515 x^{16} + 82301 x^{15} + 12498 x^{14} - 25677 x^{13} - 3560 x^{12} - 33345 x^{11} - 17836 x^{10} + 19768 x^{9} + 19004 x^{8} - 7029 x^{7} + 393 x^{6} + 406 x^{5} - 125 x^{4} + 46 x^{3} - 3 x^{2} - 3 x + 1 \)

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:  \(6856321115222930027351949403366821000578180886805057525634765625=3^{20}\cdot 5^{30}\cdot 11^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.44$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(165=3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{165}(1,·)$, $\chi_{165}(4,·)$, $\chi_{165}(133,·)$, $\chi_{165}(136,·)$, $\chi_{165}(137,·)$, $\chi_{165}(14,·)$, $\chi_{165}(16,·)$, $\chi_{165}(146,·)$, $\chi_{165}(148,·)$, $\chi_{165}(23,·)$, $\chi_{165}(152,·)$, $\chi_{165}(26,·)$, $\chi_{165}(157,·)$, $\chi_{165}(158,·)$, $\chi_{165}(31,·)$, $\chi_{165}(34,·)$, $\chi_{165}(163,·)$, $\chi_{165}(37,·)$, $\chi_{165}(38,·)$, $\chi_{165}(47,·)$, $\chi_{165}(49,·)$, $\chi_{165}(53,·)$, $\chi_{165}(56,·)$, $\chi_{165}(58,·)$, $\chi_{165}(59,·)$, $\chi_{165}(64,·)$, $\chi_{165}(67,·)$, $\chi_{165}(71,·)$, $\chi_{165}(82,·)$, $\chi_{165}(86,·)$, $\chi_{165}(89,·)$, $\chi_{165}(91,·)$, $\chi_{165}(92,·)$, $\chi_{165}(97,·)$, $\chi_{165}(103,·)$, $\chi_{165}(104,·)$, $\chi_{165}(113,·)$, $\chi_{165}(119,·)$, $\chi_{165}(122,·)$, $\chi_{165}(124,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $\frac{1}{99123971} a^{34} + \frac{48325965}{99123971} a^{33} - \frac{14706952}{99123971} a^{32} - \frac{41112703}{99123971} a^{31} + \frac{42082959}{99123971} a^{30} + \frac{20002548}{99123971} a^{29} + \frac{16925418}{99123971} a^{28} + \frac{42606392}{99123971} a^{27} + \frac{20923379}{99123971} a^{26} - \frac{46801204}{99123971} a^{25} - \frac{27232078}{99123971} a^{24} + \frac{38089811}{99123971} a^{23} - \frac{1459024}{99123971} a^{22} - \frac{43052406}{99123971} a^{21} - \frac{20263771}{99123971} a^{20} + \frac{23251585}{99123971} a^{19} + \frac{42843033}{99123971} a^{18} + \frac{14928518}{99123971} a^{17} + \frac{2564031}{99123971} a^{16} + \frac{33654159}{99123971} a^{15} + \frac{33391978}{99123971} a^{14} - \frac{6794352}{99123971} a^{13} + \frac{49046729}{99123971} a^{12} - \frac{25887237}{99123971} a^{11} - \frac{3457789}{99123971} a^{10} - \frac{3432673}{99123971} a^{9} - \frac{3537205}{99123971} a^{8} - \frac{49072231}{99123971} a^{7} + \frac{29742274}{99123971} a^{6} - \frac{4802986}{99123971} a^{5} + \frac{33699826}{99123971} a^{4} + \frac{13182749}{99123971} a^{3} + \frac{42955733}{99123971} a^{2} - \frac{32822876}{99123971} a - \frac{30707200}{99123971}$, $\frac{1}{99123971} a^{35} + \frac{11600658}{99123971} a^{33} + \frac{11516529}{99123971} a^{32} - \frac{49216346}{99123971} a^{31} - \frac{32440999}{99123971} a^{30} - \frac{23665139}{99123971} a^{29} - \frac{6978060}{99123971} a^{28} + \frac{9801477}{99123971} a^{27} - \frac{13271762}{99123971} a^{26} - \frac{12184725}{99123971} a^{25} - \frac{39991260}{99123971} a^{24} + \frac{13197898}{99123971} a^{23} + \frac{34901976}{99123971} a^{22} + \frac{35937546}{99123971} a^{21} - \frac{15929368}{99123971} a^{20} + \frac{19590626}{99123971} a^{19} + \frac{31657321}{99123971} a^{18} + \frac{29645000}{99123971} a^{17} - \frac{12630003}{99123971} a^{16} + \frac{38082044}{99123971} a^{15} + \frac{22628188}{99123971} a^{14} - \frac{27362396}{99123971} a^{13} - \frac{46751313}{99123971} a^{12} + \frac{9198667}{99123971} a^{11} - \frac{22825726}{99123971} a^{10} - \frac{4842303}{99123971} a^{9} + \frac{3585949}{99123971} a^{8} - \frac{48089634}{99123971} a^{7} + \frac{47576832}{99123971} a^{6} - \frac{20834197}{99123971} a^{5} + \frac{12119504}{99123971} a^{4} + \frac{43927151}{99123971} a^{3} + \frac{14237022}{99123971} a^{2} - \frac{22089365}{99123971} a + \frac{38046242}{99123971}$, $\frac{1}{1321619905343} a^{36} + \frac{5318}{1321619905343} a^{35} - \frac{5323}{1321619905343} a^{34} - \frac{628536600795}{1321619905343} a^{33} + \frac{224058382300}{1321619905343} a^{32} - \frac{511168454062}{1321619905343} a^{31} - \frac{2946412898}{19725670229} a^{30} + \frac{467141565411}{1321619905343} a^{29} + \frac{406653705597}{1321619905343} a^{28} - \frac{353233350186}{1321619905343} a^{27} - \frac{369407062330}{1321619905343} a^{26} + \frac{531093136513}{1321619905343} a^{25} + \frac{166372298818}{1321619905343} a^{24} + \frac{488121752500}{1321619905343} a^{23} - \frac{150130511220}{1321619905343} a^{22} - \frac{445568913610}{1321619905343} a^{21} + \frac{556245171665}{1321619905343} a^{20} + \frac{417172878655}{1321619905343} a^{19} + \frac{640163879023}{1321619905343} a^{18} + \frac{365979323404}{1321619905343} a^{17} - \frac{356459756789}{1321619905343} a^{16} - \frac{152962482811}{1321619905343} a^{15} + \frac{429984932581}{1321619905343} a^{14} - \frac{99646765276}{1321619905343} a^{13} + \frac{441246094975}{1321619905343} a^{12} + \frac{543368236805}{1321619905343} a^{11} + \frac{60896838360}{1321619905343} a^{10} + \frac{148936344274}{1321619905343} a^{9} - \frac{614631745855}{1321619905343} a^{8} - \frac{337436363481}{1321619905343} a^{7} + \frac{463293130766}{1321619905343} a^{6} + \frac{77162836580}{1321619905343} a^{5} - \frac{349388451877}{1321619905343} a^{4} + \frac{453978346043}{1321619905343} a^{3} + \frac{107307301981}{1321619905343} a^{2} - \frac{586658696779}{1321619905343} a - \frac{78263650073}{1321619905343}$, $\frac{1}{2774999210184113645874458483} a^{37} - \frac{862073583808281}{2774999210184113645874458483} a^{36} + \frac{3850064028910045540}{2774999210184113645874458483} a^{35} - \frac{3845753660991004137}{2774999210184113645874458483} a^{34} + \frac{802240245853340871760156286}{2774999210184113645874458483} a^{33} - \frac{57697937932523414721038235}{2774999210184113645874458483} a^{32} - \frac{67185680497963091664213164}{2774999210184113645874458483} a^{31} - \frac{168362435389079433271911255}{2774999210184113645874458483} a^{30} - \frac{1116160429254924668477303233}{2774999210184113645874458483} a^{29} - \frac{595249667343745116686464257}{2774999210184113645874458483} a^{28} - \frac{651823785617680357364237386}{2774999210184113645874458483} a^{27} - \frac{879486673302859121395184166}{2774999210184113645874458483} a^{26} + \frac{185317961349194762406623759}{2774999210184113645874458483} a^{25} - \frac{536394684212529111110797726}{2774999210184113645874458483} a^{24} - \frac{857298114221303268368627888}{2774999210184113645874458483} a^{23} - \frac{863766427174284891073361035}{2774999210184113645874458483} a^{22} - \frac{489178122807823161931776744}{2774999210184113645874458483} a^{21} + \frac{544478746852252756724923308}{2774999210184113645874458483} a^{20} - \frac{511565804865577614610386768}{2774999210184113645874458483} a^{19} + \frac{26079094909363782388835628}{2774999210184113645874458483} a^{18} + \frac{1358029089755204486771479429}{2774999210184113645874458483} a^{17} + \frac{1248250374352959436155268566}{2774999210184113645874458483} a^{16} + \frac{283646969771565537914979570}{2774999210184113645874458483} a^{15} - \frac{1207529602238455949641714541}{2774999210184113645874458483} a^{14} + \frac{133237936623365902906216343}{2774999210184113645874458483} a^{13} - \frac{747663814498196047907670413}{2774999210184113645874458483} a^{12} + \frac{717471959233919319398142638}{2774999210184113645874458483} a^{11} + \frac{1211889229965802882348293540}{2774999210184113645874458483} a^{10} - \frac{620735484342307203242570805}{2774999210184113645874458483} a^{9} + \frac{338466780669906132889414363}{2774999210184113645874458483} a^{8} - \frac{65263959622593560001138321}{2774999210184113645874458483} a^{7} - \frac{562435070337519115329010438}{2774999210184113645874458483} a^{6} - \frac{529944361934037348215574487}{2774999210184113645874458483} a^{5} - \frac{238206700454500586924514616}{2774999210184113645874458483} a^{4} - \frac{685602940646915257235754372}{2774999210184113645874458483} a^{3} - \frac{990108875254304473382615931}{2774999210184113645874458483} a^{2} - \frac{291889549662753537428564622}{2774999210184113645874458483} a - \frac{1248277823339672227022390724}{2774999210184113645874458483}$, $\frac{1}{2774999210184113645874458483} a^{38} - \frac{586983089948714}{2774999210184113645874458483} a^{36} + \frac{10350434840154428822}{2774999210184113645874458483} a^{35} - \frac{10347499924704685268}{2774999210184113645874458483} a^{34} + \frac{311782311713967536007985043}{2774999210184113645874458483} a^{33} + \frac{1097415633657767558813236688}{2774999210184113645874458483} a^{32} - \frac{1373319070288024418665820782}{2774999210184113645874458483} a^{31} - \frac{414172897577637557883000052}{2774999210184113645874458483} a^{30} + \frac{97419859701959644212141617}{2774999210184113645874458483} a^{29} + \frac{42789253347816627130125366}{2774999210184113645874458483} a^{28} + \frac{169251896841884666415255134}{2774999210184113645874458483} a^{27} - \frac{745178354091937811194435264}{2774999210184113645874458483} a^{26} + \frac{950570653116685112468224477}{2774999210184113645874458483} a^{25} - \frac{573946974874208291228568493}{2774999210184113645874458483} a^{24} - \frac{416349207605808926299517388}{2774999210184113645874458483} a^{23} + \frac{672817076640136887673616201}{2774999210184113645874458483} a^{22} - \frac{1138127228241409983696065130}{2774999210184113645874458483} a^{21} - \frac{607765981417398575206892063}{2774999210184113645874458483} a^{20} - \frac{831419262748862713663344191}{2774999210184113645874458483} a^{19} - \frac{1046926949883654368789171246}{2774999210184113645874458483} a^{18} + \frac{897859969907334801561660530}{2774999210184113645874458483} a^{17} - \frac{131394381686140489742546229}{2774999210184113645874458483} a^{16} - \frac{1025622387041321751195524979}{2774999210184113645874458483} a^{15} - \frac{525969630686719027203998118}{2774999210184113645874458483} a^{14} + \frac{292581908216758264163011255}{2774999210184113645874458483} a^{13} + \frac{465698027540629678796947623}{2774999210184113645874458483} a^{12} + \frac{844550317410398274740237478}{2774999210184113645874458483} a^{11} + \frac{1257579686640835672543595822}{2774999210184113645874458483} a^{10} + \frac{617684966848690003691096242}{2774999210184113645874458483} a^{9} + \frac{549164207632629425874521344}{2774999210184113645874458483} a^{8} - \frac{588385109282220378945132462}{2774999210184113645874458483} a^{7} - \frac{1136640647959676735376970354}{2774999210184113645874458483} a^{6} + \frac{288905287786925115955779251}{2774999210184113645874458483} a^{5} - \frac{711560026950951211622803561}{2774999210184113645874458483} a^{4} - \frac{952050195167801338557803803}{2774999210184113645874458483} a^{3} + \frac{946016838580830134177785158}{2774999210184113645874458483} a^{2} + \frac{785220815506885162747794773}{2774999210184113645874458483} a + \frac{1176286033850826444226481558}{2774999210184113645874458483}$, $\frac{1}{2774999210184113645874458483} a^{39} + \frac{461991690719074}{2774999210184113645874458483} a^{36} + \frac{3870965865209987824}{2774999210184113645874458483} a^{35} - \frac{6144272485748690931}{2774999210184113645874458483} a^{34} + \frac{817891095843566921449336558}{2774999210184113645874458483} a^{33} + \frac{618624222169966952640506628}{2774999210184113645874458483} a^{32} - \frac{867474609941398029383108941}{2774999210184113645874458483} a^{31} + \frac{1152278857284364403107654531}{2774999210184113645874458483} a^{30} - \frac{1109117814905858494729618544}{2774999210184113645874458483} a^{29} - \frac{217491755194649861981346854}{2774999210184113645874458483} a^{28} - \frac{1238540117027460777753560465}{2774999210184113645874458483} a^{27} - \frac{895471293545146294988807450}{2774999210184113645874458483} a^{26} - \frac{62286572269051631054283162}{2774999210184113645874458483} a^{25} - \frac{631694900963224735035240690}{2774999210184113645874458483} a^{24} + \frac{370574336998881532023892180}{2774999210184113645874458483} a^{23} + \frac{221684613623712610628274912}{2774999210184113645874458483} a^{22} - \frac{801836012745179420016097212}{2774999210184113645874458483} a^{21} + \frac{625798599415456429030551216}{2774999210184113645874458483} a^{20} - \frac{1295946947965005315535548833}{2774999210184113645874458483} a^{19} + \frac{499010771207517013643552486}{2774999210184113645874458483} a^{18} + \frac{90713700135024767826148784}{2774999210184113645874458483} a^{17} - \frac{460926638517763306301482410}{2774999210184113645874458483} a^{16} + \frac{1236263233633360144805648983}{2774999210184113645874458483} a^{15} + \frac{1084467992674805702077903877}{2774999210184113645874458483} a^{14} - \frac{1147487803713334644497788807}{2774999210184113645874458483} a^{13} - \frac{1268527255279931908661683563}{2774999210184113645874458483} a^{12} - \frac{467448611436826489034490718}{2774999210184113645874458483} a^{11} + \frac{842079897283816051871057228}{2774999210184113645874458483} a^{10} - \frac{372361084721795541503035078}{2774999210184113645874458483} a^{9} + \frac{450754708979220907613031118}{2774999210184113645874458483} a^{8} + \frac{1334201132211271819120074521}{2774999210184113645874458483} a^{7} - \frac{1057616665805869709876396367}{2774999210184113645874458483} a^{6} + \frac{1255274914840544033348103719}{2774999210184113645874458483} a^{5} - \frac{1373696503853457460554070522}{2774999210184113645874458483} a^{4} + \frac{357995671357517780452023240}{2774999210184113645874458483} a^{3} + \frac{443871784605194225329865245}{2774999210184113645874458483} a^{2} + \frac{660622981417944180768180913}{2774999210184113645874458483} a + \frac{318199573853077849982959131}{2774999210184113645874458483}$

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

Class group and class number

$C_{55}$, which has order $55$ (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:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{13490996009088925769958446}{2774999210184113645874458483} a^{39} - \frac{15484060430134754495226963}{2774999210184113645874458483} a^{38} + \frac{84404606304210684033152938}{2774999210184113645874458483} a^{37} + \frac{2617315032658843377218071}{2774999210184113645874458483} a^{36} - \frac{182109778478117088471632813}{2774999210184113645874458483} a^{35} + \frac{125111956071939158454996540}{2774999210184113645874458483} a^{34} - \frac{655599791589074153109433071}{2774999210184113645874458483} a^{33} + \frac{81174103551027305404844723}{2774999210184113645874458483} a^{32} + \frac{4022301098043340115758622450}{2774999210184113645874458483} a^{31} - \frac{3654379469424695554251041239}{2774999210184113645874458483} a^{30} - \frac{11959036725384828207095184380}{2774999210184113645874458483} a^{29} + \frac{6153328294693300335516586102}{2774999210184113645874458483} a^{28} + \frac{12183540145172556807450343827}{2774999210184113645874458483} a^{27} - \frac{26481168122912305075103486}{41417898659464382774245649} a^{26} + \frac{1050118490426881966940495731}{41417898659464382774245649} a^{25} + \frac{10007257836360429021691795851}{2774999210184113645874458483} a^{24} - \frac{336751980609239877085684124885}{2774999210184113645874458483} a^{23} - \frac{17858764456307941142987906070}{2774999210184113645874458483} a^{22} + \frac{524316469370724990562578701832}{2774999210184113645874458483} a^{21} - \frac{390158042694921921671408053743}{2774999210184113645874458483} a^{20} + \frac{17074199122645549769516522733}{2774999210184113645874458483} a^{19} + \frac{53955649691997533075279506838}{2774999210184113645874458483} a^{18} - \frac{1561070878631121119754355803008}{2774999210184113645874458483} a^{17} + \frac{1527277379033341387615926186404}{2774999210184113645874458483} a^{16} + \frac{1572235472388861662857842663275}{2774999210184113645874458483} a^{15} - \frac{2600932887348081380510947713184}{2774999210184113645874458483} a^{14} - \frac{886242395161561463914855078436}{2774999210184113645874458483} a^{13} - \frac{166129505941623639205307948981}{2774999210184113645874458483} a^{12} - \frac{652218546700143864130616001910}{2774999210184113645874458483} a^{11} - \frac{116372253068351948076618469325}{2774999210184113645874458483} a^{10} + \frac{1479286317969215989661358388640}{2774999210184113645874458483} a^{9} + \frac{357189392010001574293645574304}{2774999210184113645874458483} a^{8} - \frac{222684890015641639779385949558}{2774999210184113645874458483} a^{7} + \frac{33004492874036989649747509123}{2774999210184113645874458483} a^{6} + \frac{8057214899158645095249312649}{2774999210184113645874458483} a^{5} - \frac{425171254298231913297851321648}{2774999210184113645874458483} a^{4} + \frac{1456412497975126504085900015}{2774999210184113645874458483} a^{3} - \frac{225183999144623641765077832}{2774999210184113645874458483} a^{2} - \frac{56594616302020579466853419}{2774999210184113645874458483} a + \frac{33238379515152504937817561}{2774999210184113645874458483} \) (order $30$)
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:  \( 636238292225226.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{20}$ (as 40T2):

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 $C_2\times C_{20}$
Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{5})\), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{15})\), 10.0.52089208083.1, 10.10.669871503125.1, 10.0.162778775259375.1, 20.0.26496929674942114598525390625.1, 20.0.1402274470934209014892578125.1, 20.20.82802905234194108120391845703125.1

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 $20^{2}$ R R $20^{2}$ R $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ $20^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
3Data not computed
5Data not computed
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$