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Properties

Label 32.0.15729269092...000.17
Degree $32$
Signature $[0, 16]$
Discriminant $2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $61.29$
Ramified primes $2, 3, 5, 7$
Class number $1280$ (GRH)
Class group $[2, 2, 2, 4, 40]$ (GRH)
Galois group $C_2^3\times C_4$ (as 32T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![413596, -416736, 694048, -229664, 1478892, -12030544, 36347776, -66186296, 93434649, -137310532, 247885630, -460538212, 743691738, -999898964, 1125846650, -1078291980, 891439082, -643690432, 409713728, -231450216, 116610362, -52567076, 21235254, -7687636, 2489518, -718276, 183466, -41020, 7918, -1280, 168, -16, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 16*x^31 + 168*x^30 - 1280*x^29 + 7918*x^28 - 41020*x^27 + 183466*x^26 - 718276*x^25 + 2489518*x^24 - 7687636*x^23 + 21235254*x^22 - 52567076*x^21 + 116610362*x^20 - 231450216*x^19 + 409713728*x^18 - 643690432*x^17 + 891439082*x^16 - 1078291980*x^15 + 1125846650*x^14 - 999898964*x^13 + 743691738*x^12 - 460538212*x^11 + 247885630*x^10 - 137310532*x^9 + 93434649*x^8 - 66186296*x^7 + 36347776*x^6 - 12030544*x^5 + 1478892*x^4 - 229664*x^3 + 694048*x^2 - 416736*x + 413596)
 
gp: K = bnfinit(x^32 - 16*x^31 + 168*x^30 - 1280*x^29 + 7918*x^28 - 41020*x^27 + 183466*x^26 - 718276*x^25 + 2489518*x^24 - 7687636*x^23 + 21235254*x^22 - 52567076*x^21 + 116610362*x^20 - 231450216*x^19 + 409713728*x^18 - 643690432*x^17 + 891439082*x^16 - 1078291980*x^15 + 1125846650*x^14 - 999898964*x^13 + 743691738*x^12 - 460538212*x^11 + 247885630*x^10 - 137310532*x^9 + 93434649*x^8 - 66186296*x^7 + 36347776*x^6 - 12030544*x^5 + 1478892*x^4 - 229664*x^3 + 694048*x^2 - 416736*x + 413596, 1)
 

Normalized defining polynomial

\( x^{32} - 16 x^{31} + 168 x^{30} - 1280 x^{29} + 7918 x^{28} - 41020 x^{27} + 183466 x^{26} - 718276 x^{25} + 2489518 x^{24} - 7687636 x^{23} + 21235254 x^{22} - 52567076 x^{21} + 116610362 x^{20} - 231450216 x^{19} + 409713728 x^{18} - 643690432 x^{17} + 891439082 x^{16} - 1078291980 x^{15} + 1125846650 x^{14} - 999898964 x^{13} + 743691738 x^{12} - 460538212 x^{11} + 247885630 x^{10} - 137310532 x^{9} + 93434649 x^{8} - 66186296 x^{7} + 36347776 x^{6} - 12030544 x^{5} + 1478892 x^{4} - 229664 x^{3} + 694048 x^{2} - 416736 x + 413596 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1572926909253345615680132503044096000000000000000000000000=2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.29$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(840=2^{3}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(643,·)$, $\chi_{840}(769,·)$, $\chi_{840}(407,·)$, $\chi_{840}(281,·)$, $\chi_{840}(29,·)$, $\chi_{840}(547,·)$, $\chi_{840}(503,·)$, $\chi_{840}(421,·)$, $\chi_{840}(167,·)$, $\chi_{840}(41,·)$, $\chi_{840}(43,·)$, $\chi_{840}(307,·)$, $\chi_{840}(181,·)$, $\chi_{840}(827,·)$, $\chi_{840}(701,·)$, $\chi_{840}(449,·)$, $\chi_{840}(323,·)$, $\chi_{840}(587,·)$, $\chi_{840}(589,·)$, $\chi_{840}(461,·)$, $\chi_{840}(209,·)$, $\chi_{840}(83,·)$, $\chi_{840}(727,·)$, $\chi_{840}(601,·)$, $\chi_{840}(463,·)$, $\chi_{840}(349,·)$, $\chi_{840}(223,·)$, $\chi_{840}(743,·)$, $\chi_{840}(629,·)$, $\chi_{840}(169,·)$, $\chi_{840}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{38} a^{16} - \frac{4}{19} a^{15} + \frac{2}{19} a^{14} - \frac{1}{19} a^{13} - \frac{3}{19} a^{12} + \frac{1}{19} a^{11} - \frac{5}{19} a^{10} + \frac{2}{19} a^{8} - \frac{8}{19} a^{7} + \frac{1}{19} a^{6} + \frac{9}{19} a^{5} + \frac{1}{38} a^{4} - \frac{6}{19} a^{3} - \frac{4}{19} a^{2} - \frac{4}{19} a + \frac{8}{19}$, $\frac{1}{38} a^{17} + \frac{8}{19} a^{15} - \frac{4}{19} a^{14} + \frac{8}{19} a^{13} - \frac{4}{19} a^{12} + \frac{3}{19} a^{11} - \frac{2}{19} a^{10} + \frac{2}{19} a^{9} + \frac{8}{19} a^{8} - \frac{6}{19} a^{7} - \frac{2}{19} a^{6} - \frac{7}{38} a^{5} - \frac{2}{19} a^{4} + \frac{5}{19} a^{3} + \frac{2}{19} a^{2} - \frac{5}{19} a + \frac{7}{19}$, $\frac{1}{38} a^{18} + \frac{3}{19} a^{15} - \frac{5}{19} a^{14} - \frac{7}{19} a^{13} - \frac{6}{19} a^{12} + \frac{1}{19} a^{11} + \frac{6}{19} a^{10} + \frac{8}{19} a^{9} - \frac{7}{19} a^{7} - \frac{1}{38} a^{6} + \frac{6}{19} a^{5} - \frac{3}{19} a^{4} + \frac{3}{19} a^{3} + \frac{2}{19} a^{2} - \frac{5}{19} a + \frac{5}{19}$, $\frac{1}{38} a^{19} - \frac{1}{2} a^{7} - \frac{9}{19} a + \frac{9}{19}$, $\frac{1}{38} a^{20} - \frac{1}{2} a^{8} - \frac{9}{19} a^{2} + \frac{9}{19} a$, $\frac{1}{38} a^{21} - \frac{1}{2} a^{9} - \frac{9}{19} a^{3} + \frac{9}{19} a^{2}$, $\frac{1}{38} a^{22} - \frac{1}{2} a^{10} - \frac{9}{19} a^{4} + \frac{9}{19} a^{3}$, $\frac{1}{38} a^{23} - \frac{1}{2} a^{11} - \frac{9}{19} a^{5} + \frac{9}{19} a^{4}$, $\frac{1}{50996} a^{24} - \frac{3}{12749} a^{23} + \frac{111}{25498} a^{22} - \frac{27}{2318} a^{21} + \frac{3}{50996} a^{20} - \frac{3}{671} a^{19} - \frac{1}{122} a^{18} + \frac{112}{12749} a^{17} - \frac{455}{50996} a^{16} + \frac{3082}{12749} a^{15} - \frac{5495}{25498} a^{14} + \frac{371}{1159} a^{13} + \frac{20711}{50996} a^{12} + \frac{3266}{12749} a^{11} + \frac{17}{418} a^{10} + \frac{5399}{25498} a^{9} + \frac{25051}{50996} a^{8} - \frac{5851}{12749} a^{7} + \frac{113}{2318} a^{6} - \frac{4806}{12749} a^{5} - \frac{16567}{50996} a^{4} - \frac{5129}{12749} a^{3} + \frac{2891}{25498} a^{2} - \frac{3550}{12749} a - \frac{5235}{25498}$, $\frac{1}{50996} a^{25} + \frac{39}{25498} a^{23} - \frac{307}{25498} a^{22} - \frac{415}{50996} a^{21} - \frac{48}{12749} a^{20} - \frac{235}{25498} a^{19} - \frac{271}{25498} a^{18} - \frac{447}{50996} a^{17} + \frac{79}{25498} a^{16} - \frac{12047}{25498} a^{15} + \frac{3319}{12749} a^{14} - \frac{14225}{50996} a^{13} - \frac{2372}{12749} a^{12} + \frac{9637}{25498} a^{11} + \frac{9791}{25498} a^{10} + \frac{1735}{4636} a^{9} + \frac{2873}{12749} a^{8} - \frac{4981}{25498} a^{7} + \frac{7317}{25498} a^{6} + \frac{23829}{50996} a^{5} + \frac{11791}{25498} a^{4} - \frac{767}{25498} a^{3} + \frac{267}{671} a^{2} - \frac{4547}{25498} a - \frac{3899}{12749}$, $\frac{1}{968924} a^{26} + \frac{3}{484462} a^{25} + \frac{3}{968924} a^{24} + \frac{524}{242231} a^{23} + \frac{3407}{968924} a^{22} - \frac{269}{242231} a^{21} + \frac{2179}{968924} a^{20} + \frac{1757}{242231} a^{19} + \frac{3495}{968924} a^{18} + \frac{277}{22021} a^{17} + \frac{6953}{968924} a^{16} - \frac{35161}{242231} a^{15} - \frac{342275}{968924} a^{14} - \frac{227445}{484462} a^{13} + \frac{121413}{968924} a^{12} - \frac{31651}{242231} a^{11} - \frac{13571}{50996} a^{10} + \frac{4405}{22021} a^{9} + \frac{237451}{968924} a^{8} + \frac{32657}{242231} a^{7} + \frac{105011}{968924} a^{6} + \frac{80680}{242231} a^{5} + \frac{103557}{968924} a^{4} + \frac{59328}{242231} a^{3} + \frac{21073}{242231} a^{2} - \frac{4413}{22021} a + \frac{11679}{484462}$, $\frac{1}{968924} a^{27} + \frac{5}{968924} a^{25} + \frac{7}{968924} a^{24} - \frac{6851}{968924} a^{23} + \frac{1337}{242231} a^{22} - \frac{865}{968924} a^{21} + \frac{5943}{968924} a^{20} + \frac{7687}{968924} a^{19} - \frac{2567}{484462} a^{18} + \frac{8951}{968924} a^{17} + \frac{53}{50996} a^{16} + \frac{14971}{88084} a^{15} - \frac{113247}{484462} a^{14} + \frac{1143}{15884} a^{13} + \frac{207417}{968924} a^{12} - \frac{144871}{968924} a^{11} + \frac{95651}{242231} a^{10} + \frac{28331}{968924} a^{9} - \frac{237621}{968924} a^{8} + \frac{88083}{968924} a^{7} + \frac{10677}{484462} a^{6} - \frac{381277}{968924} a^{5} + \frac{196059}{968924} a^{4} + \frac{8999}{22021} a^{3} - \frac{214625}{484462} a^{2} - \frac{122295}{484462} a + \frac{225091}{484462}$, $\frac{1}{10658164} a^{28} - \frac{3}{10658164} a^{27} - \frac{5}{10658164} a^{26} + \frac{103}{10658164} a^{25} + \frac{71}{10658164} a^{24} + \frac{87591}{10658164} a^{23} - \frac{86851}{10658164} a^{22} - \frac{10359}{968924} a^{21} + \frac{32649}{10658164} a^{20} - \frac{34825}{10658164} a^{19} + \frac{6115}{968924} a^{18} - \frac{109023}{10658164} a^{17} - \frac{4073}{968924} a^{16} - \frac{237535}{10658164} a^{15} - \frac{20623}{10658164} a^{14} - \frac{423965}{968924} a^{13} - \frac{2586649}{10658164} a^{12} - \frac{1437937}{10658164} a^{11} - \frac{4392611}{10658164} a^{10} + \frac{262139}{10658164} a^{9} + \frac{2833201}{10658164} a^{8} + \frac{4081855}{10658164} a^{7} - \frac{4478867}{10658164} a^{6} + \frac{3787857}{10658164} a^{5} + \frac{582504}{2664541} a^{4} + \frac{603409}{2664541} a^{3} + \frac{1085669}{5329082} a^{2} + \frac{3677}{5329082} a - \frac{841287}{2664541}$, $\frac{1}{841994956} a^{29} + \frac{25}{841994956} a^{28} + \frac{1}{11078881} a^{27} - \frac{389}{841994956} a^{26} + \frac{6057}{841994956} a^{25} + \frac{997}{210498739} a^{24} + \frac{2757163}{420997478} a^{23} - \frac{8593401}{841994956} a^{22} - \frac{21201}{44315524} a^{21} - \frac{2385769}{210498739} a^{20} + \frac{3753885}{420997478} a^{19} + \frac{10137433}{841994956} a^{18} + \frac{8151021}{841994956} a^{17} - \frac{1119935}{210498739} a^{16} + \frac{4220644}{210498739} a^{15} - \frac{3719901}{13803196} a^{14} + \frac{404815269}{841994956} a^{13} + \frac{4185761}{38272498} a^{12} + \frac{143299219}{420997478} a^{11} - \frac{13804095}{44315524} a^{10} + \frac{63219361}{841994956} a^{9} + \frac{69698522}{210498739} a^{8} - \frac{181221647}{420997478} a^{7} - \frac{3001945}{44315524} a^{6} - \frac{79990731}{210498739} a^{5} - \frac{2508965}{841994956} a^{4} - \frac{78973859}{210498739} a^{3} - \frac{5367700}{11078881} a^{2} + \frac{47297703}{210498739} a - \frac{105984127}{420997478}$, $\frac{1}{659255448399384561257095890650444} a^{30} - \frac{15}{659255448399384561257095890650444} a^{29} + \frac{4613252790083516724392365}{659255448399384561257095890650444} a^{28} - \frac{64585539061169234141492095}{659255448399384561257095890650444} a^{27} + \frac{47795581760316719061610742}{164813862099846140314273972662611} a^{26} + \frac{58767444706451400276068959}{29966156745426570966231631393202} a^{25} + \frac{711525727120882652149649369}{329627724199692280628547945325222} a^{24} - \frac{119503346978781725928040145681}{659255448399384561257095890650444} a^{23} - \frac{3966268746574896717465565175353}{329627724199692280628547945325222} a^{22} + \frac{1474614744184661650606300586}{148347310620923618644711046501} a^{21} + \frac{568440281991862805034231592040}{164813862099846140314273972662611} a^{20} - \frac{2673675460925872201319109036165}{659255448399384561257095890650444} a^{19} - \frac{842029832229384147160947883841}{164813862099846140314273972662611} a^{18} - \frac{1653606959894262571748492441837}{164813862099846140314273972662611} a^{17} - \frac{120471391037797355383593147023}{329627724199692280628547945325222} a^{16} - \frac{86777456343649755882164079335927}{659255448399384561257095890650444} a^{15} + \frac{15595830660126413672631777653749}{164813862099846140314273972662611} a^{14} - \frac{21475158350380422175465908384574}{164813862099846140314273972662611} a^{13} + \frac{51473682719763917565676434419462}{164813862099846140314273972662611} a^{12} - \frac{141411625012798388162929768425929}{659255448399384561257095890650444} a^{11} - \frac{39515717697468919281790654507817}{329627724199692280628547945325222} a^{10} - \frac{68565391185715213867369505756757}{164813862099846140314273972662611} a^{9} - \frac{14250104586724615528566897533553}{164813862099846140314273972662611} a^{8} - \frac{122046812391517021543298585677453}{659255448399384561257095890650444} a^{7} - \frac{18896524888358622206569987559729}{659255448399384561257095890650444} a^{6} + \frac{5472512182658674737724935562129}{21266304787076921330874060988724} a^{5} - \frac{184381365403039185325297061852811}{659255448399384561257095890650444} a^{4} - \frac{31755942771403833865626411374441}{164813862099846140314273972662611} a^{3} - \frac{79600858829755341751938110826171}{164813862099846140314273972662611} a^{2} - \frac{14480803906960855010811055245205}{29966156745426570966231631393202} a - \frac{145690355691502271964661662299575}{329627724199692280628547945325222}$, $\frac{1}{317285802950207402257253853156355037876} a^{31} + \frac{60156}{79321450737551850564313463289088759469} a^{30} + \frac{172012593419972458073472432643}{317285802950207402257253853156355037876} a^{29} + \frac{13500585169693104161165289553431}{317285802950207402257253853156355037876} a^{28} + \frac{86546557186233883394250293487203}{317285802950207402257253853156355037876} a^{27} + \frac{30601863289521790000094218020475}{317285802950207402257253853156355037876} a^{26} + \frac{328229425656724243407588841783151}{79321450737551850564313463289088759469} a^{25} + \frac{2350226523399592813776574645574955}{317285802950207402257253853156355037876} a^{24} + \frac{2937450348241995736262442810073487331}{317285802950207402257253853156355037876} a^{23} - \frac{331264980976779930712632918389420283}{28844163904564309296113986650577730716} a^{22} + \frac{1663928828715766241312637812881566723}{158642901475103701128626926578177518938} a^{21} + \frac{538195916311494234871561094453370151}{317285802950207402257253853156355037876} a^{20} - \frac{1513546595089492454380766443757549753}{317285802950207402257253853156355037876} a^{19} + \frac{1489245877657508426196906224349696243}{317285802950207402257253853156355037876} a^{18} - \frac{183391494260110938495878903354033115}{79321450737551850564313463289088759469} a^{17} - \frac{280493784407666302970116476571191735}{28844163904564309296113986650577730716} a^{16} + \frac{106396736033362355467534933907240617617}{317285802950207402257253853156355037876} a^{15} - \frac{37702696088078400763462591084330451531}{317285802950207402257253853156355037876} a^{14} + \frac{137536137816155941697278688546571925}{14422081952282154648056993325288865358} a^{13} + \frac{118788549035631222380989978003495572059}{317285802950207402257253853156355037876} a^{12} + \frac{67705840448186701308951057911249405}{285585781233309993030831550995819116} a^{11} + \frac{35457512666482122957585930622947787199}{317285802950207402257253853156355037876} a^{10} - \frac{17883519033261076345611674853936369389}{158642901475103701128626926578177518938} a^{9} + \frac{60004902820388851472412146481320725823}{317285802950207402257253853156355037876} a^{8} - \frac{43874961411332750861300302877673056171}{158642901475103701128626926578177518938} a^{7} - \frac{145699156085569067481993822821238297277}{317285802950207402257253853156355037876} a^{6} - \frac{104325242527069730024181095606172207807}{317285802950207402257253853156355037876} a^{5} + \frac{8642434137866327205851121956292278074}{79321450737551850564313463289088759469} a^{4} + \frac{69835122703276503492350576059468172975}{158642901475103701128626926578177518938} a^{3} - \frac{50298351691607535459892178944462327011}{158642901475103701128626926578177518938} a^{2} + \frac{66344823796168091003396718871588440039}{158642901475103701128626926578177518938} a + \frac{22531328327351375757108942876871141178}{79321450737551850564313463289088759469}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{40}$, which has order $1280$ (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:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{7798263178585677}{6015169293686627720602} a^{30} - \frac{116973947678785155}{6015169293686627720602} a^{29} + \frac{2370312738261656455}{12030338587373255441204} a^{28} - \frac{4338476020783566515}{3007584646843313860301} a^{27} + \frac{103708890776579490455}{12030338587373255441204} a^{26} - \frac{129557184996691806458}{3007584646843313860301} a^{25} + \frac{1118164573191367195675}{6015169293686627720602} a^{24} - \frac{2107648987348469210900}{3007584646843313860301} a^{23} + \frac{28076971432302696421445}{12030338587373255441204} a^{22} - \frac{18685819392180411230}{2707096891848167291} a^{21} + \frac{54689256255348501553273}{3007584646843313860301} a^{20} - \frac{128411213634651368820280}{3007584646843313860301} a^{19} + \frac{1073020703482701254225155}{12030338587373255441204} a^{18} - \frac{45163392996722478152710}{273416786076664896391} a^{17} + \frac{810553120736404526257225}{3007584646843313860301} a^{16} - \frac{1154361741442419359975256}{3007584646843313860301} a^{15} + \frac{514916564506944898747085}{1093667144306659585564} a^{14} - \frac{24040341139167306291270}{49304666341693669841} a^{13} + \frac{1243921464508723915728255}{3007584646843313860301} a^{12} - \frac{822230632443088927658720}{3007584646843313860301} a^{11} + \frac{1552687431256661261790905}{12030338587373255441204} a^{10} - \frac{117237510471297894869320}{3007584646843313860301} a^{9} + \frac{43697194517692103365830}{3007584646843313860301} a^{8} - \frac{5987169716153811956960}{273416786076664896391} a^{7} + \frac{276004594631566367604525}{12030338587373255441204} a^{6} - \frac{5353408105549308052855}{546833572153329792782} a^{5} - \frac{31764325079637847281595}{12030338587373255441204} a^{4} + \frac{16178010001350522868635}{3007584646843313860301} a^{3} - \frac{5530854097677050535985}{3007584646843313860301} a^{2} - \frac{817303850569565718130}{3007584646843313860301} a + \frac{5302022785688877100981}{6015169293686627720602} \) (order $6$)
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:  \( 18539393548258.133 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3\times C_4$ (as 32T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2^3\times C_4$
Character table for $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{210}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{2}, \sqrt{-35})\), \(\Q(\sqrt{-6}, \sqrt{-35})\), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{-6}, \sqrt{-70})\), \(\Q(\sqrt{-3}, \sqrt{-70})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{105})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{10}, \sqrt{-14})\), \(\Q(\sqrt{-30}, \sqrt{-35})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{5}, \sqrt{-14})\), \(\Q(\sqrt{-7}, \sqrt{10})\), \(\Q(\sqrt{21}, \sqrt{-30})\), \(\Q(\sqrt{-15}, \sqrt{42})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), \(\Q(\sqrt{-6}, \sqrt{-14})\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{42})\), \(\Q(\sqrt{-7}, \sqrt{-30})\), \(\Q(\sqrt{-14}, \sqrt{-15})\), \(\Q(\sqrt{10}, \sqrt{21})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\sqrt{-14}, \sqrt{-30})\), \(\Q(\sqrt{10}, \sqrt{42})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{-14})\), \(\Q(\sqrt{-3}, \sqrt{10})\), 4.0.98000.1, \(\Q(\zeta_{20})^+\), 4.4.8000.1, 4.0.392000.2, 4.4.3528000.1, 4.0.72000.2, 4.0.18000.1, 4.4.882000.1, 8.0.497871360000.14, 8.0.6146560000.2, 8.0.497871360000.1, 8.0.497871360000.9, 8.0.497871360000.5, 8.0.121550625.1, 8.0.497871360000.11, 8.0.497871360000.6, 8.0.497871360000.7, 8.0.497871360000.19, 8.0.497871360000.16, 8.0.207360000.1, 8.0.796594176.2, 8.8.497871360000.1, 8.0.497871360000.10, 8.0.9604000000.2, 8.0.153664000000.4, 8.0.12446784000000.10, 8.0.777924000000.4, 8.0.2458624000000.2, 8.0.2458624000000.4, 8.0.199148544000000.17, 8.0.199148544000000.16, 8.0.2458624000000.6, \(\Q(\zeta_{40})^+\), 8.8.199148544000000.8, 8.0.82944000000.6, 8.0.199148544000000.67, 8.0.82944000000.2, 8.0.82944000000.3, 8.0.199148544000000.193, 8.0.199148544000000.140, 8.8.199148544000000.7, 8.8.199148544000000.4, 8.0.199148544000000.220, 8.0.777924000000.1, 8.8.777924000000.2, 8.8.12446784000000.2, 8.0.12446784000000.1, 8.0.777924000000.9, 8.0.324000000.2, 8.0.5184000000.4, 8.0.12446784000000.17, 16.0.247875891108249600000000.1, 16.0.6044831973376000000000000.2, 16.0.39660142577319936000000000000.69, 16.0.39660142577319936000000000000.70, 16.0.39660142577319936000000000000.68, 16.0.605165749776000000000000.10, 16.0.154922431942656000000000000.15, 16.0.39660142577319936000000000000.5, 16.0.39660142577319936000000000000.56, 16.0.39660142577319936000000000000.23, 16.0.39660142577319936000000000000.43, 16.0.39660142577319936000000000000.52, 16.0.6879707136000000000000.5, 16.0.39660142577319936000000000000.32, 16.16.39660142577319936000000000000.5

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 R ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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
$2$2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$