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Properties

Label 32.0.56078041364...0321.1
Degree $32$
Signature $[0, 16]$
Discriminant $3^{16}\cdot 11^{16}\cdot 17^{28}$
Root discriminant $68.53$
Ramified primes $3, 11, 17$
Class number Not computed
Class group Not computed
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![23670130201, -12778402507, -17053058431, 12988964210, 4210930219, -5684845246, -87681207, 1472291823, -163392606, -307629712, 31923685, 74949056, -18829429, -9364589, 7358296, -2098612, 1298951, -336490, -649829, 448827, -109778, -19516, 39027, -17726, 4169, 369, -1228, 506, -61, -8, 12, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 6*x^31 + 12*x^30 - 8*x^29 - 61*x^28 + 506*x^27 - 1228*x^26 + 369*x^25 + 4169*x^24 - 17726*x^23 + 39027*x^22 - 19516*x^21 - 109778*x^20 + 448827*x^19 - 649829*x^18 - 336490*x^17 + 1298951*x^16 - 2098612*x^15 + 7358296*x^14 - 9364589*x^13 - 18829429*x^12 + 74949056*x^11 + 31923685*x^10 - 307629712*x^9 - 163392606*x^8 + 1472291823*x^7 - 87681207*x^6 - 5684845246*x^5 + 4210930219*x^4 + 12988964210*x^3 - 17053058431*x^2 - 12778402507*x + 23670130201)
 
gp: K = bnfinit(x^32 - 6*x^31 + 12*x^30 - 8*x^29 - 61*x^28 + 506*x^27 - 1228*x^26 + 369*x^25 + 4169*x^24 - 17726*x^23 + 39027*x^22 - 19516*x^21 - 109778*x^20 + 448827*x^19 - 649829*x^18 - 336490*x^17 + 1298951*x^16 - 2098612*x^15 + 7358296*x^14 - 9364589*x^13 - 18829429*x^12 + 74949056*x^11 + 31923685*x^10 - 307629712*x^9 - 163392606*x^8 + 1472291823*x^7 - 87681207*x^6 - 5684845246*x^5 + 4210930219*x^4 + 12988964210*x^3 - 17053058431*x^2 - 12778402507*x + 23670130201, 1)
 

Normalized defining polynomial

\( x^{32} - 6 x^{31} + 12 x^{30} - 8 x^{29} - 61 x^{28} + 506 x^{27} - 1228 x^{26} + 369 x^{25} + 4169 x^{24} - 17726 x^{23} + 39027 x^{22} - 19516 x^{21} - 109778 x^{20} + 448827 x^{19} - 649829 x^{18} - 336490 x^{17} + 1298951 x^{16} - 2098612 x^{15} + 7358296 x^{14} - 9364589 x^{13} - 18829429 x^{12} + 74949056 x^{11} + 31923685 x^{10} - 307629712 x^{9} - 163392606 x^{8} + 1472291823 x^{7} - 87681207 x^{6} - 5684845246 x^{5} + 4210930219 x^{4} + 12988964210 x^{3} - 17053058431 x^{2} - 12778402507 x + 23670130201 \)

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:  \(56078041364540796715130003864299185186303615574764488400321=3^{16}\cdot 11^{16}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.53$
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:  $3, 11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(561=3\cdot 11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{561}(1,·)$, $\chi_{561}(518,·)$, $\chi_{561}(263,·)$, $\chi_{561}(395,·)$, $\chi_{561}(529,·)$, $\chi_{561}(274,·)$, $\chi_{561}(406,·)$, $\chi_{561}(155,·)$, $\chi_{561}(287,·)$, $\chi_{561}(32,·)$, $\chi_{561}(166,·)$, $\chi_{561}(298,·)$, $\chi_{561}(43,·)$, $\chi_{561}(560,·)$, $\chi_{561}(307,·)$, $\chi_{561}(188,·)$, $\chi_{561}(67,·)$, $\chi_{561}(331,·)$, $\chi_{561}(76,·)$, $\chi_{561}(461,·)$, $\chi_{561}(463,·)$, $\chi_{561}(208,·)$, $\chi_{561}(472,·)$, $\chi_{561}(89,·)$, $\chi_{561}(353,·)$, $\chi_{561}(98,·)$, $\chi_{561}(100,·)$, $\chi_{561}(485,·)$, $\chi_{561}(230,·)$, $\chi_{561}(494,·)$, $\chi_{561}(373,·)$, $\chi_{561}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{8}$, $\frac{1}{268} a^{24} + \frac{9}{67} a^{23} + \frac{27}{134} a^{22} - \frac{15}{67} a^{21} + \frac{13}{67} a^{20} - \frac{11}{134} a^{19} + \frac{27}{268} a^{18} - \frac{13}{67} a^{17} + \frac{23}{134} a^{16} - \frac{15}{67} a^{15} + \frac{23}{134} a^{14} + \frac{27}{134} a^{13} + \frac{59}{268} a^{12} + \frac{49}{134} a^{11} + \frac{29}{134} a^{10} + \frac{37}{134} a^{9} - \frac{7}{134} a^{8} + \frac{19}{67} a^{7} - \frac{93}{268} a^{6} + \frac{59}{134} a^{5} - \frac{15}{67} a^{4} - \frac{8}{67} a^{3} + \frac{37}{134} a^{2} - \frac{2}{67} a + \frac{131}{268}$, $\frac{1}{268} a^{25} - \frac{9}{67} a^{23} + \frac{3}{134} a^{22} - \frac{33}{134} a^{21} - \frac{9}{134} a^{20} + \frac{15}{268} a^{19} + \frac{12}{67} a^{18} + \frac{21}{134} a^{17} + \frac{13}{134} a^{16} + \frac{31}{134} a^{15} + \frac{3}{134} a^{14} - \frac{9}{268} a^{13} - \frac{4}{67} a^{12} + \frac{7}{134} a^{11} + \frac{65}{134} a^{10} - \frac{33}{67} a^{9} - \frac{45}{134} a^{8} - \frac{15}{268} a^{7} - \frac{9}{134} a^{6} - \frac{5}{67} a^{5} - \frac{4}{67} a^{4} + \frac{5}{67} a^{3} + \frac{2}{67} a^{2} + \frac{17}{268} a - \frac{13}{134}$, $\frac{1}{268} a^{26} - \frac{19}{134} a^{23} + \frac{1}{134} a^{22} - \frac{17}{134} a^{21} + \frac{11}{268} a^{20} + \frac{15}{67} a^{19} - \frac{29}{134} a^{18} + \frac{15}{134} a^{17} - \frac{6}{67} a^{16} - \frac{5}{134} a^{15} + \frac{39}{268} a^{14} + \frac{13}{67} a^{13} - \frac{3}{134} a^{12} - \frac{47}{134} a^{11} + \frac{20}{67} a^{10} - \frac{53}{134} a^{9} + \frac{17}{268} a^{8} + \frac{19}{134} a^{7} + \frac{29}{67} a^{6} - \frac{14}{67} a^{5} + \frac{1}{67} a^{4} - \frac{18}{67} a^{3} + \frac{1}{268} a^{2} + \frac{22}{67} a - \frac{27}{67}$, $\frac{1}{27604} a^{27} + \frac{1}{27604} a^{26} - \frac{19}{27604} a^{25} - \frac{23}{13802} a^{24} - \frac{3371}{13802} a^{23} - \frac{557}{13802} a^{22} + \frac{2381}{27604} a^{21} + \frac{3213}{27604} a^{20} - \frac{2921}{27604} a^{19} - \frac{423}{6901} a^{18} - \frac{2667}{13802} a^{17} + \frac{245}{6901} a^{16} - \frac{6833}{27604} a^{15} + \frac{949}{27604} a^{14} - \frac{4637}{27604} a^{13} - \frac{3}{206} a^{12} + \frac{2883}{6901} a^{11} - \frac{2351}{13802} a^{10} - \frac{6883}{27604} a^{9} + \frac{403}{27604} a^{8} + \frac{2377}{27604} a^{7} + \frac{3923}{13802} a^{6} + \frac{2191}{6901} a^{5} + \frac{3261}{6901} a^{4} + \frac{13339}{27604} a^{3} + \frac{10199}{27604} a^{2} + \frac{123}{27604} a + \frac{1610}{6901}$, $\frac{1}{55208} a^{28} - \frac{1}{55208} a^{27} + \frac{41}{27604} a^{26} - \frac{1}{6901} a^{25} + \frac{45}{55208} a^{24} - \frac{6381}{27604} a^{23} + \frac{7493}{55208} a^{22} - \frac{6493}{55208} a^{21} + \frac{4339}{27604} a^{20} + \frac{265}{13802} a^{19} + \frac{7217}{55208} a^{18} + \frac{1456}{6901} a^{17} - \frac{6939}{55208} a^{16} + \frac{12143}{55208} a^{15} - \frac{5997}{27604} a^{14} - \frac{5349}{27604} a^{13} - \frac{7337}{55208} a^{12} - \frac{607}{13802} a^{11} - \frac{14989}{55208} a^{10} + \frac{27}{824} a^{9} - \frac{1899}{13802} a^{8} + \frac{9477}{27604} a^{7} - \frac{24129}{55208} a^{6} - \frac{1739}{13802} a^{5} + \frac{23}{55208} a^{4} - \frac{17303}{55208} a^{3} + \frac{2995}{27604} a^{2} + \frac{8453}{27604} a - \frac{16485}{55208}$, $\frac{1}{55208} a^{29} - \frac{1}{55208} a^{27} - \frac{1}{6901} a^{26} - \frac{53}{55208} a^{25} - \frac{87}{55208} a^{24} - \frac{12745}{55208} a^{23} + \frac{1089}{6901} a^{22} - \frac{8481}{55208} a^{21} - \frac{2971}{13802} a^{20} + \frac{599}{55208} a^{19} - \frac{13577}{55208} a^{18} - \frac{4923}{55208} a^{17} + \frac{1678}{6901} a^{16} + \frac{9611}{55208} a^{15} - \frac{3287}{27604} a^{14} - \frac{361}{55208} a^{13} - \frac{7495}{55208} a^{12} + \frac{19991}{55208} a^{11} - \frac{2274}{6901} a^{10} + \frac{23843}{55208} a^{9} - \frac{3123}{6901} a^{8} - \frac{26637}{55208} a^{7} - \frac{5575}{55208} a^{6} + \frac{21375}{55208} a^{5} + \frac{1129}{6901} a^{4} - \frac{13723}{55208} a^{3} - \frac{2745}{27604} a^{2} - \frac{25733}{55208} a - \frac{3815}{55208}$, $\frac{1}{50199236596588769652218913950488} a^{30} - \frac{47399578628080938486983985}{25099618298294384826109456975244} a^{29} + \frac{57455261011723495637558799}{50199236596588769652218913950488} a^{28} - \frac{112362388359399129458204394}{6274904574573596206527364243811} a^{27} + \frac{4369399549331951916409913385}{50199236596588769652218913950488} a^{26} + \frac{85347763003120437288046776971}{50199236596588769652218913950488} a^{25} + \frac{61016346210599670965122158735}{50199236596588769652218913950488} a^{24} + \frac{5378319272480607289389482197589}{25099618298294384826109456975244} a^{23} - \frac{7592040390011851523663900560453}{50199236596588769652218913950488} a^{22} - \frac{884034270190005835064167947561}{6274904574573596206527364243811} a^{21} - \frac{781557115777230615498716139283}{3861479738199136127093762611576} a^{20} - \frac{9387215741050858932680455218795}{50199236596588769652218913950488} a^{19} - \frac{8661629143991784335839507820383}{50199236596588769652218913950488} a^{18} + \frac{1839927195623407408209299291311}{25099618298294384826109456975244} a^{17} + \frac{10605663514838564034180780212567}{50199236596588769652218913950488} a^{16} - \frac{3962667905129897424086034414321}{25099618298294384826109456975244} a^{15} + \frac{10297483042769260073490041133009}{50199236596588769652218913950488} a^{14} - \frac{2790215171313176755216947693461}{50199236596588769652218913950488} a^{13} + \frac{3924934232553464318914399342431}{50199236596588769652218913950488} a^{12} - \frac{6196638002669505238395999556271}{25099618298294384826109456975244} a^{11} - \frac{20501652031121478587046080895453}{50199236596588769652218913950488} a^{10} - \frac{2950990281825813866824890062427}{6274904574573596206527364243811} a^{9} + \frac{20005539749127242280632355273713}{50199236596588769652218913950488} a^{8} + \frac{1487261934682811787413879005143}{50199236596588769652218913950488} a^{7} - \frac{593714146567920114477146820685}{50199236596588769652218913950488} a^{6} + \frac{6012731928899302976338168967795}{25099618298294384826109456975244} a^{5} + \frac{17228394683564223410505651516849}{50199236596588769652218913950488} a^{4} + \frac{2003371534537284914199126570545}{25099618298294384826109456975244} a^{3} + \frac{1745626609790109077058275470245}{50199236596588769652218913950488} a^{2} + \frac{214733193438345232873938790591}{50199236596588769652218913950488} a + \frac{5181709985519714227562600891215}{12549809149147192413054728487622}$, $\frac{1}{999944300525586012606225799772669180667055493893495434300645389473255428732178075925488531469941390772104} a^{31} - \frac{2013739028131499848855197807735455414379393966177316012412630215114771621}{999944300525586012606225799772669180667055493893495434300645389473255428732178075925488531469941390772104} a^{30} + \frac{1110996264676052510511110475777919813945281815020817659654846893316521206470334908023263328173135473}{249986075131396503151556449943167295166763873473373858575161347368313857183044518981372132867485347693026} a^{29} - \frac{6097718753858436128427480551532626988512757197134394386559975564261207403063468060164286893139867575}{999944300525586012606225799772669180667055493893495434300645389473255428732178075925488531469941390772104} a^{28} + \frac{1264177480859290844512421739056727254929439772126177764359008429892716396137886571136740602624331}{432128046899561803200616162390954702103308337896929746888783660100801827455565287781109996313717109236} a^{27} + \frac{218432179161390552837028648710313585237750229575353856405987687777097856223940803944787419651943338002}{124993037565698251575778224971583647583381936736686929287580673684156928591522259490686066433742673846513} a^{26} + \frac{17900808573188660384960951895421954658087425706108894209216171214433660859094508119017342426147961463}{9708197092481417598118697085171545443369470814499955672821799897798596395458039572092121664756712531768} a^{25} + \frac{42602755466453918578427750313957264578233605850797483597810421640568583517280240516166019908416277959}{249986075131396503151556449943167295166763873473373858575161347368313857183044518981372132867485347693026} a^{24} + \frac{93913884087661517763595423894681901808886701307691847987388031114012853501804804695799824525155596331943}{499972150262793006303112899886334590333527746946747717150322694736627714366089037962744265734970695386052} a^{23} + \frac{188780300639281899464760776473286568910649820815193663611925597243065604631921622387082165727670285165485}{999944300525586012606225799772669180667055493893495434300645389473255428732178075925488531469941390772104} a^{22} - \frac{46287376551336344761856696996336331859624319035642538095584757359643775052820861770097029563531950820609}{499972150262793006303112899886334590333527746946747717150322694736627714366089037962744265734970695386052} a^{21} + \frac{103026000035467110500782581032501591510128892024408027976693912697762428287246427143036585772912578678629}{499972150262793006303112899886334590333527746946747717150322694736627714366089037962744265734970695386052} a^{20} - \frac{5536183744862855880785636041768195989037562317199669302320361520345021676408106425821363749690242225015}{76918792348122000969709676905589936974388884145653494946203491497942725287090621225037579343841645444008} a^{19} - \frac{3250086091309629328205146664430568137302971716249091780556034399004544787567349406313249284355292868873}{249986075131396503151556449943167295166763873473373858575161347368313857183044518981372132867485347693026} a^{18} + \frac{10705938250592907075424955884978193569927830390083023082666455762588422547823711966852671310876897085711}{249986075131396503151556449943167295166763873473373858575161347368313857183044518981372132867485347693026} a^{17} - \frac{50487179262981369437142437691947111519387572009913271765250259624163219022530095808851825398434323752693}{999944300525586012606225799772669180667055493893495434300645389473255428732178075925488531469941390772104} a^{16} - \frac{11583333814325782264266939357170778417166833098424354327979689943168639857892012094893050879030229092376}{124993037565698251575778224971583647583381936736686929287580673684156928591522259490686066433742673846513} a^{15} + \frac{7261762100837146681391297180574661841864851465823460670612364548397383304828893831159010563045963876349}{499972150262793006303112899886334590333527746946747717150322694736627714366089037962744265734970695386052} a^{14} + \frac{487240044796343719417840699555054846466134640743992125310830926309490285900285454271152214500762575071}{9708197092481417598118697085171545443369470814499955672821799897798596395458039572092121664756712531768} a^{13} + \frac{9005890963635309615967577269694320253260327997557804344683557557755635490490373966788103988187566478099}{249986075131396503151556449943167295166763873473373858575161347368313857183044518981372132867485347693026} a^{12} - \frac{179464283760664818803488061266146475152944648098821360800515309006613208898767185006606133827693778833499}{499972150262793006303112899886334590333527746946747717150322694736627714366089037962744265734970695386052} a^{11} - \frac{250616260144640793569684360981705560987038308142720503990903892038705685643771348701143215689846417923031}{999944300525586012606225799772669180667055493893495434300645389473255428732178075925488531469941390772104} a^{10} - \frac{58174689327503405037832113310211752533161064322477618675586866967627822852180631849234305917265589033239}{499972150262793006303112899886334590333527746946747717150322694736627714366089037962744265734970695386052} a^{9} - \frac{29559461359954132318357573295492730431670674855982281532609683270363519942514544138850838121940404274401}{124993037565698251575778224971583647583381936736686929287580673684156928591522259490686066433742673846513} a^{8} + \frac{469344444118686666245005057999916161438966306779324203418888224266502558290690427727963020367022067260189}{999944300525586012606225799772669180667055493893495434300645389473255428732178075925488531469941390772104} a^{7} + \frac{78273897980748162674568083820564941657867821954913664088488892045564757467554003693842474408057968544651}{249986075131396503151556449943167295166763873473373858575161347368313857183044518981372132867485347693026} a^{6} + \frac{42267211332043911174310342009381975914216475276866429867225550428592389475282433318963143089996077140840}{124993037565698251575778224971583647583381936736686929287580673684156928591522259490686066433742673846513} a^{5} + \frac{232082664576625964307563081846540589512137545368524715837188257490671617205507366546360209030640970188029}{999944300525586012606225799772669180667055493893495434300645389473255428732178075925488531469941390772104} a^{4} + \frac{111698492556708564522441230481740376488373323851550752135635634835916056582860259730944183721714679716395}{499972150262793006303112899886334590333527746946747717150322694736627714366089037962744265734970695386052} a^{3} + \frac{101854893886795110601036625433254705471854794990818262536338099691931419399593306978451736363243481367745}{499972150262793006303112899886334590333527746946747717150322694736627714366089037962744265734970695386052} a^{2} + \frac{117153360573092665760402750583688599665197818290956278408995033318820062143735994822517771729007277801889}{499972150262793006303112899886334590333527746946747717150322694736627714366089037962744265734970695386052} a - \frac{249738388682206956105147057327869834824935445252504702942133806967749090450725270049547769318856607}{6499433221269839082009384402913657894112196176128172285527200924747030755290365846991495222455111704}$

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:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{117066152004660062459749717276243872525083822038585192592666460869633041}{1472520291359527762530507331112319272560101202878678717864804236669642283409569896} a^{31} + \frac{117168597280753095502141991327843919042900516308963078764733102480023265}{736260145679763881265253665556159636280050601439339358932402118334821141704784948} a^{30} - \frac{953701393274569563876711442413778725662778316142240370916462966421402253}{368130072839881940632626832778079818140025300719669679466201059167410570852392474} a^{29} + \frac{2673874090247418727966518612770576200779455217292138067174302277173886613}{368130072839881940632626832778079818140025300719669679466201059167410570852392474} a^{28} - \frac{138680893601826778061714405157847333087462088345537722064054847728946863}{8272585906514200913092737815237748722247759566734150100364068745335069007918932} a^{27} + \frac{29475630678491203056159552389053384056426336698787562181665247793607922503}{1472520291359527762530507331112319272560101202878678717864804236669642283409569896} a^{26} + \frac{1277851928884349425353375337998053763440578566719587114123512265042993965}{7148156754172464866652948209283103264854860208148925814877690469270108171891116} a^{25} - \frac{1013039849219534302538558723621279940981345161364490717777883541144667761815}{1472520291359527762530507331112319272560101202878678717864804236669642283409569896} a^{24} + \frac{779982970883698927339734596428964008521959343320515669583273146816328591735}{736260145679763881265253665556159636280050601439339358932402118334821141704784948} a^{23} - \frac{233239022286273959805197015631624237689000835948485593042981679152433805355}{368130072839881940632626832778079818140025300719669679466201059167410570852392474} a^{22} - \frac{3924700308361566213447751023119336548568860436745636996776870667889306436911}{736260145679763881265253665556159636280050601439339358932402118334821141704784948} a^{21} + \frac{31424646124488059781670851103520698062879716522047412146156149711291584395905}{1472520291359527762530507331112319272560101202878678717864804236669642283409569896} a^{20} - \frac{13221905903765524096698240037870869541435543088803592814568152713791823646379}{368130072839881940632626832778079818140025300719669679466201059167410570852392474} a^{19} + \frac{2836758643694962114502098504044494230921282423306838516222284034304091110667}{113270791643040597117731333162486097889238554067590670604984941282280175646889992} a^{18} + \frac{24811371374232152081673917938977630071667464538030283467897762632880873345864}{184065036419940970316313416389039909070012650359834839733100529583705285426196237} a^{17} - \frac{269489126188930361544254569635664868155329471793888477776491806538064474451869}{736260145679763881265253665556159636280050601439339358932402118334821141704784948} a^{16} + \frac{202605514550962729975947960998950047717292231348165966362177172263261154369435}{736260145679763881265253665556159636280050601439339358932402118334821141704784948} a^{15} - \frac{536953157906139783681044175271612019312222156462992951112541047159654647144643}{1472520291359527762530507331112319272560101202878678717864804236669642283409569896} a^{14} + \frac{16670307575285245431992182284216847876482570211285062300265167443187754327}{7148156754172464866652948209283103264854860208148925814877690469270108171891116} a^{13} + \frac{4008954683887276735268593336244968834077169213631416547687793021982281029688673}{1472520291359527762530507331112319272560101202878678717864804236669642283409569896} a^{12} - \frac{420124327649903888793139470694291512070185955858668444327369348437647922002159}{56635395821520298558865666581243048944619277033795335302492470641140087823444996} a^{11} + \frac{641327257888495722227358332603023978791457428329756484597871275189500823257415}{368130072839881940632626832778079818140025300719669679466201059167410570852392474} a^{10} + \frac{21211827297474300675046230460545328101409977623783463871200431241650648418022203}{736260145679763881265253665556159636280050601439339358932402118334821141704784948} a^{9} + \frac{969114542075932125082228053292746409892438001046352537381730971874050044631639}{1472520291359527762530507331112319272560101202878678717864804236669642283409569896} a^{8} - \frac{93344421880759066183111950388720860856400213279878063371155630069004634081820235}{736260145679763881265253665556159636280050601439339358932402118334821141704784948} a^{7} + \frac{36750888327189780203713153140436652462038124392936808219362813044076970025694029}{1472520291359527762530507331112319272560101202878678717864804236669642283409569896} a^{6} + \frac{210745152677690022974207864075486843088845816805270015079119382053679516398651281}{368130072839881940632626832778079818140025300719669679466201059167410570852392474} a^{5} - \frac{344673567265347686516373396136853907636660731125782829947632313724407205674862081}{736260145679763881265253665556159636280050601439339358932402118334821141704784948} a^{4} - \frac{307909442937990400338908922027095289919452025222007359419794011511282540830162328}{184065036419940970316313416389039909070012650359834839733100529583705285426196237} a^{3} + \frac{3913182749769545590465422706424630366138572007692596528622425540738173729571787749}{1472520291359527762530507331112319272560101202878678717864804236669642283409569896} a^{2} + \frac{3282126851476553160218812076992878420179603620752705582975404677965516913436279527}{1472520291359527762530507331112319272560101202878678717864804236669642283409569896} a - \frac{46391590287335939364661397143733511322212330795919316066602264739991590070767}{9571080404804179124805866267442650828139571422211612000343216727025773530296} \) (order $6$)
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

$C_2^2\times C_8$ (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{-187}) \), \(\Q(\sqrt{561}) \), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\sqrt{-3}, \sqrt{-187})\), \(\Q(\sqrt{-11}, \sqrt{17})\), \(\Q(\sqrt{-11}, \sqrt{-51})\), \(\Q(\sqrt{17}, \sqrt{33})\), \(\Q(\sqrt{33}, \sqrt{-51})\), 4.4.4913.1, 4.0.44217.1, 4.0.594473.1, 4.4.5350257.1, 8.0.99049307841.1, 8.0.1955143089.1, 8.0.28625249966049.3, 8.0.353398147729.1, 8.0.28625249966049.1, 8.8.28625249966049.1, 8.0.28625249966049.2, 8.0.33237432513.1, \(\Q(\zeta_{17})^+\), 8.8.486629249422833.1, 8.0.6007768511393.1, 16.0.819404935618788275652670401.1, 16.0.1104726920056229495169.1, 16.0.236808026393829811663621745889.3, 16.0.236808026393829811663621745889.1, 16.0.36093282486485263170800449.1, 16.0.236808026393829811663621745889.2, 16.16.236808026393829811663621745889.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$ 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
3Data not computed
11Data not computed
17Data not computed