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Properties

Label 32.0.12176701499...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{48}\cdot 5^{16}\cdot 17^{28}$
Root discriminant $75.45$
Ramified primes $2, 5, 17$
Class number $1133568$ (GRH)
Class group $[12, 48, 1968]$ (GRH)
Galois group $C_2^2\times C_8$ (as 32T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3492717761, 342344184, 7334969884, 460160520, 7725637406, 339375200, 5372660276, 132064062, 2730786641, 3685932, 1066070798, -26993986, 328619726, -18587306, 81158763, -7017350, 16372434, -1863692, 2686108, -342938, 393657, -44612, 40562, -6664, 6534, -478, -85, -204, 160, 18, -11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - 11*x^30 + 18*x^29 + 160*x^28 - 204*x^27 - 85*x^26 - 478*x^25 + 6534*x^24 - 6664*x^23 + 40562*x^22 - 44612*x^21 + 393657*x^20 - 342938*x^19 + 2686108*x^18 - 1863692*x^17 + 16372434*x^16 - 7017350*x^15 + 81158763*x^14 - 18587306*x^13 + 328619726*x^12 - 26993986*x^11 + 1066070798*x^10 + 3685932*x^9 + 2730786641*x^8 + 132064062*x^7 + 5372660276*x^6 + 339375200*x^5 + 7725637406*x^4 + 460160520*x^3 + 7334969884*x^2 + 342344184*x + 3492717761)
 
gp: K = bnfinit(x^32 - 2*x^31 - 11*x^30 + 18*x^29 + 160*x^28 - 204*x^27 - 85*x^26 - 478*x^25 + 6534*x^24 - 6664*x^23 + 40562*x^22 - 44612*x^21 + 393657*x^20 - 342938*x^19 + 2686108*x^18 - 1863692*x^17 + 16372434*x^16 - 7017350*x^15 + 81158763*x^14 - 18587306*x^13 + 328619726*x^12 - 26993986*x^11 + 1066070798*x^10 + 3685932*x^9 + 2730786641*x^8 + 132064062*x^7 + 5372660276*x^6 + 339375200*x^5 + 7725637406*x^4 + 460160520*x^3 + 7334969884*x^2 + 342344184*x + 3492717761, 1)
 

Normalized defining polynomial

\( x^{32} - 2 x^{31} - 11 x^{30} + 18 x^{29} + 160 x^{28} - 204 x^{27} - 85 x^{26} - 478 x^{25} + 6534 x^{24} - 6664 x^{23} + 40562 x^{22} - 44612 x^{21} + 393657 x^{20} - 342938 x^{19} + 2686108 x^{18} - 1863692 x^{17} + 16372434 x^{16} - 7017350 x^{15} + 81158763 x^{14} - 18587306 x^{13} + 328619726 x^{12} - 26993986 x^{11} + 1066070798 x^{10} + 3685932 x^{9} + 2730786641 x^{8} + 132064062 x^{7} + 5372660276 x^{6} + 339375200 x^{5} + 7725637406 x^{4} + 460160520 x^{3} + 7334969884 x^{2} + 342344184 x + 3492717761 \)

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:  \(1217670149940184118315109788118094613693071360000000000000000=2^{48}\cdot 5^{16}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.45$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(680=2^{3}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{680}(1,·)$, $\chi_{680}(259,·)$, $\chi_{680}(9,·)$, $\chi_{680}(529,·)$, $\chi_{680}(659,·)$, $\chi_{680}(409,·)$, $\chi_{680}(281,·)$, $\chi_{680}(161,·)$, $\chi_{680}(291,·)$, $\chi_{680}(169,·)$, $\chi_{680}(171,·)$, $\chi_{680}(49,·)$, $\chi_{680}(179,·)$, $\chi_{680}(569,·)$, $\chi_{680}(59,·)$, $\chi_{680}(321,·)$, $\chi_{680}(579,·)$, $\chi_{680}(331,·)$, $\chi_{680}(531,·)$, $\chi_{680}(81,·)$, $\chi_{680}(339,·)$, $\chi_{680}(441,·)$, $\chi_{680}(89,·)$, $\chi_{680}(219,·)$, $\chi_{680}(611,·)$, $\chi_{680}(361,·)$, $\chi_{680}(491,·)$, $\chi_{680}(451,·)$, $\chi_{680}(19,·)$, $\chi_{680}(489,·)$, $\chi_{680}(121,·)$, $\chi_{680}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{4} a^{24} - \frac{1}{2} a^{21} - \frac{1}{4} a^{18} - \frac{1}{2} a^{16} + \frac{1}{4} a^{12} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4}$, $\frac{1}{4} a^{25} - \frac{1}{2} a^{22} - \frac{1}{4} a^{19} - \frac{1}{2} a^{17} + \frac{1}{4} a^{13} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a$, $\frac{1}{4} a^{26} - \frac{1}{2} a^{23} - \frac{1}{4} a^{20} - \frac{1}{2} a^{18} + \frac{1}{4} a^{14} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{27} - \frac{1}{4} a^{21} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} + \frac{1}{4} a^{15} - \frac{1}{2} a^{12} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{28} - \frac{1}{4} a^{22} - \frac{1}{2} a^{20} - \frac{1}{2} a^{19} + \frac{1}{4} a^{16} - \frac{1}{2} a^{13} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} + \frac{1}{4} a^{4} - \frac{1}{2} a$, $\frac{1}{52} a^{29} - \frac{5}{52} a^{28} - \frac{1}{26} a^{27} + \frac{1}{26} a^{26} - \frac{3}{52} a^{25} + \frac{1}{52} a^{24} - \frac{5}{52} a^{23} - \frac{1}{52} a^{22} + \frac{1}{26} a^{21} + \frac{9}{26} a^{20} + \frac{1}{52} a^{19} + \frac{23}{52} a^{18} + \frac{7}{52} a^{17} + \frac{25}{52} a^{16} + \frac{11}{26} a^{15} - \frac{5}{13} a^{14} + \frac{19}{52} a^{13} + \frac{9}{52} a^{12} - \frac{1}{52} a^{11} + \frac{17}{52} a^{10} - \frac{1}{13} a^{8} - \frac{1}{52} a^{7} - \frac{9}{52} a^{6} - \frac{21}{52} a^{5} - \frac{1}{4} a^{4} - \frac{2}{13} a^{3} - \frac{5}{13} a^{2} + \frac{23}{52} a + \frac{1}{4}$, $\frac{1}{4766970676} a^{30} - \frac{720663}{2383485338} a^{29} - \frac{185338607}{4766970676} a^{28} - \frac{30948457}{1191742669} a^{27} - \frac{355249855}{4766970676} a^{26} - \frac{72941490}{1191742669} a^{25} + \frac{593482003}{4766970676} a^{24} - \frac{509697558}{1191742669} a^{23} + \frac{179317381}{366690052} a^{22} + \frac{754008643}{2383485338} a^{21} + \frac{1131553005}{4766970676} a^{20} + \frac{958999307}{2383485338} a^{19} - \frac{935595361}{4766970676} a^{18} + \frac{46788363}{183345026} a^{17} - \frac{1469949695}{4766970676} a^{16} - \frac{18661111}{2383485338} a^{15} + \frac{1375172401}{4766970676} a^{14} + \frac{530607913}{2383485338} a^{13} - \frac{1294974833}{4766970676} a^{12} - \frac{559906437}{2383485338} a^{11} - \frac{71537575}{4766970676} a^{10} - \frac{82186781}{1191742669} a^{9} - \frac{1182632811}{4766970676} a^{8} - \frac{366313096}{1191742669} a^{7} - \frac{39747693}{366690052} a^{6} - \frac{166253728}{1191742669} a^{5} - \frac{923190159}{4766970676} a^{4} - \frac{50680037}{2383485338} a^{3} - \frac{1057977171}{4766970676} a^{2} + \frac{505046825}{2383485338} a - \frac{7187042}{91672513}$, $\frac{1}{13658283432568583131920802877387955013095019607131224576716143872381843778677372953370499844361532127180242814515409531167826299209536108} a^{31} + \frac{1272640976119832809805980772241665731353760845434777336845352474564691879617365248415898710395843396488719208075815998804429961}{13658283432568583131920802877387955013095019607131224576716143872381843778677372953370499844361532127180242814515409531167826299209536108} a^{30} - \frac{58909470301687363659731344435321325044809835736205814277655200438136062784963831389605946112778127861904415164082906693946768158478473}{13658283432568583131920802877387955013095019607131224576716143872381843778677372953370499844361532127180242814515409531167826299209536108} a^{29} - \frac{131344036428876726369299340810490064835516132565724393224462463543818747416526255379495149503644634183263063133384929273527836802599619}{13658283432568583131920802877387955013095019607131224576716143872381843778677372953370499844361532127180242814515409531167826299209536108} a^{28} - \frac{209855762170744814748221969805522344216435027238282770678977210352277018233907343205002682133875940786197401757929175592778537367883535}{13658283432568583131920802877387955013095019607131224576716143872381843778677372953370499844361532127180242814515409531167826299209536108} a^{27} - \frac{193016925615365841701342200548111675009792769707556323100967125861098265448265131847093390518946261241419527394169969965940361614298004}{3414570858142145782980200719346988753273754901782806144179035968095460944669343238342624961090383031795060703628852382791956574802384027} a^{26} + \frac{113071925688780418512337321152713571068310077406378439710184951864714292683303256797056749649271455874019694963332181624554721019136639}{3414570858142145782980200719346988753273754901782806144179035968095460944669343238342624961090383031795060703628852382791956574802384027} a^{25} - \frac{943608770062081143831724466995076497165738603278403180916601183279344331549463913354077828300224087633502072421840162587530498392265497}{13658283432568583131920802877387955013095019607131224576716143872381843778677372953370499844361532127180242814515409531167826299209536108} a^{24} - \frac{216878506662090223150914045015481645539362251366395525139304119858694857546740445507217682860103742677779797287329070736296343942008737}{1050637187120660240916984836722150385622693815933171121285857220952449521436720996413115372643194779013864831885800733166755869169964316} a^{23} - \frac{2251502452934885949551104790310128899141023043501369150868266855247868358081267923276821418780074640249155141829073681185856210620269271}{13658283432568583131920802877387955013095019607131224576716143872381843778677372953370499844361532127180242814515409531167826299209536108} a^{22} + \frac{231306711059656764648006856040971929160149473723794951354481732565405524270200480966809536339735372321206099394002462533936980986371983}{13658283432568583131920802877387955013095019607131224576716143872381843778677372953370499844361532127180242814515409531167826299209536108} a^{21} - \frac{405034284857245330366634395361484954447860203829122780356753327128921738127088153260048115600442825314113774709553544185369218023971418}{3414570858142145782980200719346988753273754901782806144179035968095460944669343238342624961090383031795060703628852382791956574802384027} a^{20} + \frac{783411542026110814366545836703931739626101596738745972357655039584796688323996965798210533126612742582866941779227603656949710730662592}{3414570858142145782980200719346988753273754901782806144179035968095460944669343238342624961090383031795060703628852382791956574802384027} a^{19} + \frac{495147571278162548918770695749901974985551543981941097869458444766751095440105270652266583761981055736914752246116814659322782310229175}{1050637187120660240916984836722150385622693815933171121285857220952449521436720996413115372643194779013864831885800733166755869169964316} a^{18} - \frac{3189665307513217818901008890503935388461772658532378348372661693078879371224896347916250344177283332371447782248329195837290007639935011}{13658283432568583131920802877387955013095019607131224576716143872381843778677372953370499844361532127180242814515409531167826299209536108} a^{17} - \frac{3462967539916870469597458518633844789238523522841223739490401204900616264913602067391598236392742315331860632180070679166339943327189101}{13658283432568583131920802877387955013095019607131224576716143872381843778677372953370499844361532127180242814515409531167826299209536108} a^{16} + \frac{573005092709578578769922601204241661164572885359530428001289839564447418299068823487994838539557982644227750143590011802191780785640571}{13658283432568583131920802877387955013095019607131224576716143872381843778677372953370499844361532127180242814515409531167826299209536108} a^{15} - \frac{1224323467088899360681290383125042401370891409511634176523089370168277566772749391533584727370878081483259010464091011464683830222586701}{6829141716284291565960401438693977506547509803565612288358071936190921889338686476685249922180766063590121407257704765583913149604768054} a^{14} - \frac{417204118220650513666579256956923749338199812986771091069194046370751247459355213528469391928600760534960481196513981597470253630747385}{6829141716284291565960401438693977506547509803565612288358071936190921889338686476685249922180766063590121407257704765583913149604768054} a^{13} - \frac{2743182234127510799833705087831486376343883904680331711944990296802592218922538017408161582643399979164904160219106484520177544122838099}{13658283432568583131920802877387955013095019607131224576716143872381843778677372953370499844361532127180242814515409531167826299209536108} a^{12} + \frac{5176592219848039271673852254586624082478227191196686890529658107014399923588891039045908871755567542816902704836133261226537397424901743}{13658283432568583131920802877387955013095019607131224576716143872381843778677372953370499844361532127180242814515409531167826299209536108} a^{11} - \frac{3615378928454753924571913722018847541670966726326384266780828697864358599189489259479777025675732307574998509998438968242837637504689143}{13658283432568583131920802877387955013095019607131224576716143872381843778677372953370499844361532127180242814515409531167826299209536108} a^{10} - \frac{1961101234460607825029847198326610453447709722880554255214910058109827448493364624702156616379843536816666881670216720013832674432194891}{13658283432568583131920802877387955013095019607131224576716143872381843778677372953370499844361532127180242814515409531167826299209536108} a^{9} + \frac{1831011887375870759705565574242697493077023731867900713715423223008696245004006974302831768576042078575950830568541352740548796271030367}{6829141716284291565960401438693977506547509803565612288358071936190921889338686476685249922180766063590121407257704765583913149604768054} a^{8} - \frac{108687450064088953438880979635761746563469749923701688048375656306231057182609564127529602710322172397417601369110747248210145589603419}{525318593560330120458492418361075192811346907966585560642928610476224760718360498206557686321597389506932415942900366583377934584982158} a^{7} - \frac{1633109174549160171898342978000952005999673473295306007185028204199321081745467069653978319151278271884375720188827800446968285257960311}{13658283432568583131920802877387955013095019607131224576716143872381843778677372953370499844361532127180242814515409531167826299209536108} a^{6} + \frac{6539023580540684332135030345098386320868245847836813400692512713175318618658039868969390775962363175033612504761691652378218881512139045}{13658283432568583131920802877387955013095019607131224576716143872381843778677372953370499844361532127180242814515409531167826299209536108} a^{5} + \frac{3972109480847378886796693089587827325900265924622321969080603525284733115539513027298776417893290201417504690644322026055881453563458901}{13658283432568583131920802877387955013095019607131224576716143872381843778677372953370499844361532127180242814515409531167826299209536108} a^{4} - \frac{3623937430631864311468094540151536081188412754699907891653388165287909697634407248732661212864651315707354010117725831328788415134635}{290601775161033683657889422923147979002021693768749459079066890901741356993135594752563826475777279301707293925859777258889921259777364} a^{3} + \frac{230245350976581334960547693456380518738622186186894733720032926950955764465315327340956182754745489841892503223682223808953966108435331}{6829141716284291565960401438693977506547509803565612288358071936190921889338686476685249922180766063590121407257704765583913149604768054} a^{2} - \frac{200700564585562801832377174738843547603788437404973735941979660805643671292425280159516970722275030994469899121777377948748156772594633}{1050637187120660240916984836722150385622693815933171121285857220952449521436720996413115372643194779013864831885800733166755869169964316} a + \frac{39489979597908344624039052580312306485663066752541743374624399662662957815569307391755809924012443309949860314749970170896025996163375}{525318593560330120458492418361075192811346907966585560642928610476224760718360498206557686321597389506932415942900366583377934584982158}$

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

Class group and class number

$C_{12}\times C_{48}\times C_{1968}$, which has order $1133568$ (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:  \( -1 \) (order $2$)
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:  \( 42597284566.379654 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_8$ (as 32T37):

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 $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{-170}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-2}, \sqrt{85})\), \(\Q(\sqrt{-10}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{-34})\), \(\Q(\sqrt{-2}, \sqrt{17})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{-10}, \sqrt{-34})\), 4.4.4913.1, 4.0.7860800.1, 4.0.314432.2, 4.4.122825.1, 8.0.213813760000.1, 8.0.61792176640000.44, 8.0.61792176640000.6, 8.0.98867482624.1, 8.0.61792176640000.40, 8.8.15085980625.1, 8.0.61792176640000.52, 8.0.1050467002880000.6, \(\Q(\zeta_{17})^+\), 8.8.256461670625.1, 8.0.1680747204608.1, 16.0.3818273093908961689600000000.1, 16.0.1103480924139689928294400000000.4, 16.0.1103480924139689928294400000000.1, 16.0.1103480924139689928294400000000.3, 16.0.2824911165797606216433664.1, 16.0.1103480924139689928294400000000.2, 16.16.65772588499765987890625.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
5Data not computed
17Data not computed