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Properties

Label 34.0.21463951011...2336.1
Degree $34$
Signature $[0, 17]$
Discriminant $-\,2^{51}\cdot 17^{65}$
Root discriminant $636.61$
Ramified primes $2, 17$
Class number Not computed
Class group Not computed
Galois group $C_{34}$ (as 34T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9260317344744472576, 0, 64999230103116185600, 0, 149963841244491972608, 0, 148124589852441526272, 0, 74485344198610567168, 0, 21891500132012650496, 0, 4102155135962263552, 0, 517593118432397312, 0, 45517202334845952, 0, 2848934734898432, 0, 128302687873920, 0, 4166986405120, 0, 96986208224, 0, 1591395840, 0, 17850952, 0, 129472, 0, 544, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^34 + 544*x^32 + 129472*x^30 + 17850952*x^28 + 1591395840*x^26 + 96986208224*x^24 + 4166986405120*x^22 + 128302687873920*x^20 + 2848934734898432*x^18 + 45517202334845952*x^16 + 517593118432397312*x^14 + 4102155135962263552*x^12 + 21891500132012650496*x^10 + 74485344198610567168*x^8 + 148124589852441526272*x^6 + 149963841244491972608*x^4 + 64999230103116185600*x^2 + 9260317344744472576)
 
gp: K = bnfinit(x^34 + 544*x^32 + 129472*x^30 + 17850952*x^28 + 1591395840*x^26 + 96986208224*x^24 + 4166986405120*x^22 + 128302687873920*x^20 + 2848934734898432*x^18 + 45517202334845952*x^16 + 517593118432397312*x^14 + 4102155135962263552*x^12 + 21891500132012650496*x^10 + 74485344198610567168*x^8 + 148124589852441526272*x^6 + 149963841244491972608*x^4 + 64999230103116185600*x^2 + 9260317344744472576, 1)
 

Normalized defining polynomial

\( x^{34} + 544 x^{32} + 129472 x^{30} + 17850952 x^{28} + 1591395840 x^{26} + 96986208224 x^{24} + 4166986405120 x^{22} + 128302687873920 x^{20} + 2848934734898432 x^{18} + 45517202334845952 x^{16} + 517593118432397312 x^{14} + 4102155135962263552 x^{12} + 21891500132012650496 x^{10} + 74485344198610567168 x^{8} + 148124589852441526272 x^{6} + 149963841244491972608 x^{4} + 64999230103116185600 x^{2} + 9260317344744472576 \)

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

Invariants

Degree:  $34$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 17]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-214639510116448210748631066635120907555924504566182022171002423006964754343412080865644260622336=-\,2^{51}\cdot 17^{65}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $636.61$
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:  $2, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2312=2^{3}\cdot 17^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{2312}(1,·)$, $\chi_{2312}(1155,·)$, $\chi_{2312}(2177,·)$, $\chi_{2312}(137,·)$, $\chi_{2312}(1291,·)$, $\chi_{2312}(273,·)$, $\chi_{2312}(1427,·)$, $\chi_{2312}(409,·)$, $\chi_{2312}(1563,·)$, $\chi_{2312}(545,·)$, $\chi_{2312}(1699,·)$, $\chi_{2312}(681,·)$, $\chi_{2312}(1835,·)$, $\chi_{2312}(817,·)$, $\chi_{2312}(1971,·)$, $\chi_{2312}(953,·)$, $\chi_{2312}(2107,·)$, $\chi_{2312}(1089,·)$, $\chi_{2312}(2243,·)$, $\chi_{2312}(1225,·)$, $\chi_{2312}(203,·)$, $\chi_{2312}(1361,·)$, $\chi_{2312}(339,·)$, $\chi_{2312}(1497,·)$, $\chi_{2312}(475,·)$, $\chi_{2312}(1633,·)$, $\chi_{2312}(67,·)$, $\chi_{2312}(611,·)$, $\chi_{2312}(1769,·)$, $\chi_{2312}(747,·)$, $\chi_{2312}(1905,·)$, $\chi_{2312}(883,·)$, $\chi_{2312}(2041,·)$, $\chi_{2312}(1019,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{4096} a^{24}$, $\frac{1}{4096} a^{25}$, $\frac{1}{8192} a^{26}$, $\frac{1}{8192} a^{27}$, $\frac{1}{16384} a^{28}$, $\frac{1}{16384} a^{29}$, $\frac{1}{32768} a^{30}$, $\frac{1}{32768} a^{31}$, $\frac{1}{117894786786446123263066961778324620943241964010832870686689469385328639336188787617543806707635491473442313512222317209541869568} a^{32} + \frac{5081294724258556770057308055827779368408110958216196393147931565771428512583129764114416463738925561622429041641230716801}{29473696696611530815766740444581155235810491002708217671672367346332159834047196904385951676908872868360578378055579302385467392} a^{30} - \frac{631506805810923347536062692732275524857577227156607637867672484725179248279907857720464971841921177870645017831723287683969}{29473696696611530815766740444581155235810491002708217671672367346332159834047196904385951676908872868360578378055579302385467392} a^{28} + \frac{300058519749014077201809175196315905685474786745414773251861631689906073346193179642904873524337171055927831762715363390395}{7368424174152882703941685111145288808952622750677054417918091836583039958511799226096487919227218217090144594513894825596366848} a^{26} + \frac{391069839314206105140321757814989469902188973917536541565794335334371825043450860441118486496467314789114638455432963882573}{3684212087076441351970842555572644404476311375338527208959045918291519979255899613048243959613609108545072297256947412798183424} a^{24} - \frac{22186683063720935825454290182181539937286890551961161801113977361061554489018308627200755474776631827081895739688108274347}{921053021769110337992710638893161101119077843834631802239761479572879994813974903262060989903402277136268074314236853199545856} a^{22} + \frac{378694199470552671581296114156875370465213975198534517946581190576728631516951357323363825433243999402045336518882929419467}{921053021769110337992710638893161101119077843834631802239761479572879994813974903262060989903402277136268074314236853199545856} a^{20} - \frac{308811612559756244809182631339972117490855132716082480812394797459861787658485204721995526561400703660138573704258522450929}{460526510884555168996355319446580550559538921917315901119880739786439997406987451631030494951701138568134037157118426599772928} a^{18} + \frac{439249490801702646023693183690577073142412155613862538767934765764301329198126537617043705788068405631625198846253466670673}{460526510884555168996355319446580550559538921917315901119880739786439997406987451631030494951701138568134037157118426599772928} a^{16} - \frac{169899484992429693966706829620622865874288900856531989837902318252306687663522645218317316849041915902377433535367474021591}{230263255442277584498177659723290275279769460958657950559940369893219998703493725815515247475850569284067018578559213299886464} a^{14} + \frac{13567857890465191409660000993039258170940475925627486657790696452602282207168277189654799703730081493677918080941890445671}{7195726732571174515568051866352821102492795654958060954998136559163124959484178931734851483620330290127094330579975415621452} a^{12} + \frac{55166264478407280761779244426081395447718972207352881926892785205618427983077697894364550706341823735708835962865910916291}{57565813860569396124544414930822568819942365239664487639985092473304999675873431453878811868962642321016754644639803324971616} a^{10} - \frac{290110058029898450865002799866946732087655507411990369852160267636802248769880296969425808145490713635987960228980309342345}{14391453465142349031136103732705642204985591309916121909996273118326249918968357863469702967240660580254188661159950831242904} a^{8} - \frac{118076407648032867610956104615642291826029631151425582858949951071477474568953292035237720511375727016700384301551199719801}{3597863366285587257784025933176410551246397827479030477499068279581562479742089465867425741810165145063547165289987707810726} a^{6} + \frac{4487058514712340488087927960022512880910622580062316578821083279577092046404838549656754857048793783432103679311413372505}{7195726732571174515568051866352821102492795654958060954998136559163124959484178931734851483620330290127094330579975415621452} a^{4} - \frac{26250259990980414634613513949559605191228987022180958595928468099170901893868571171623703808009408140652252965350749682713}{3597863366285587257784025933176410551246397827479030477499068279581562479742089465867425741810165145063547165289987707810726} a^{2} - \frac{277687828081372510725889666299858679769479157404448128015626583008382556854037396478248631728436362621011266264490902140192}{1798931683142793628892012966588205275623198913739515238749534139790781239871044732933712870905082572531773582644993853905363}$, $\frac{1}{240341137606356572006951149750025020527239670526231966011979959115216661451229815758638127160832666366199814422211001419592522094411776} a^{33} + \frac{1238577066528184861564666711701361561359060551318274334918118288280794550662750421104251375837078614243984957311014866653281196643}{120170568803178286003475574875012510263619835263115983005989979557608330725614907879319063580416333183099907211105500709796261047205888} a^{31} + \frac{199371392969727293873599906042122784166567769017887770849414215343076574262306107923429866512207804597613802012729259661338173231}{15021321100397285750434446859376563782952479407889497875748747444701041340701863484914882947552041647887488401388187588724532630900736} a^{29} + \frac{300623371491878277925479272681663356335849702347624615992685401030348724463403373603198268187563683198100802870801975629196412051}{15021321100397285750434446859376563782952479407889497875748747444701041340701863484914882947552041647887488401388187588724532630900736} a^{27} - \frac{442133842286263932728458238554424924832097434034269179782363936946883715638720846830655773441081077879254394933517554004480614115}{7510660550198642875217223429688281891476239703944748937874373722350520670350931742457441473776020823943744200694093794362266315450368} a^{25} - \frac{1478597806003971765072584411457970879124411255516395707545290549508286059514665624458174062498907323030140500884565402086722013341}{7510660550198642875217223429688281891476239703944748937874373722350520670350931742457441473776020823943744200694093794362266315450368} a^{23} + \frac{1803485536621226980672081813998868336926542160618140348887845103860972860646421381055482744908994203010850305673341371115456312687}{3755330275099321437608611714844140945738119851972374468937186861175260335175465871228720736888010411971872100347046897181133157725184} a^{21} + \frac{87471999221170488929937279902552517438481490153560700685336316582858112467215785373932960493560906847719394500034651519960097729}{469416284387415179701076464355517618217264981496546808617148357646907541896933233903590092111001301496484012543380862147641644715648} a^{19} - \frac{857752893543777886324749920784746782694549960851344802844992863061608997785347917658687159736576790081103475562316708838517560101}{469416284387415179701076464355517618217264981496546808617148357646907541896933233903590092111001301496484012543380862147641644715648} a^{17} + \frac{114016697335639797790159513148894692264600784380313696577757701687895099601664867053860650606852647457227967017076987867116877883}{117354071096853794925269116088879404554316245374136702154287089411726885474233308475897523027750325374121003135845215536910411178912} a^{15} + \frac{258997808302843674575532337398619566775429552255434037508273203359241389084469238027100477639612712896420936860836995198563957935}{234708142193707589850538232177758809108632490748273404308574178823453770948466616951795046055500650748242006271690431073820822357824} a^{13} - \frac{52736636249841899889857185191820349323903143239478931861635618782578978223777483367288104646652954365703608339052651140660887435}{3667314721776681091414659877777481392322382667941771942321471544116465171069790889871797594617197667941281347995162985528450349341} a^{11} - \frac{128822020090290744937976987791322993575379201926491974123256290174019701483037673556640654470182725662321848193593728371703856357}{7334629443553362182829319755554962784644765335883543884642943088232930342139581779743595189234395335882562695990325971056900698682} a^{9} + \frac{232939947710386413385101364061468909087604954912671197809616817743277609710947271424828576990036820776267280969150662603297924343}{7334629443553362182829319755554962784644765335883543884642943088232930342139581779743595189234395335882562695990325971056900698682} a^{7} - \frac{1280306870132400282144620708284764087459620278793445373416681866604574891964089974203016635417876030152359602430669026488449096047}{14669258887106724365658639511109925569289530671767087769285886176465860684279163559487190378468790671765125391980651942113801397364} a^{5} - \frac{1301253661388233476852194429969159059271094879084637891099956616877850127469142359346706509367153476685121498111343979585486198037}{7334629443553362182829319755554962784644765335883543884642943088232930342139581779743595189234395335882562695990325971056900698682} a^{3} - \frac{1033610655939499873931365724089284816868225546147556935661209096792032394594023155194386948771694117053371236618017738135452653739}{3667314721776681091414659877777481392322382667941771942321471544116465171069790889871797594617197667941281347995162985528450349341} a$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $16$
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:  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

$C_{34}$ (as 34T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 34
The 34 conjugacy class representatives for $C_{34}$
Character table for $C_{34}$ is not computed

Intermediate fields

\(\Q(\sqrt{-34}) \), 17.17.2367911594760467245844106297320951247361.1

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 $34$ $17^{2}$ $17^{2}$ $34$ $34$ R $17^{2}$ $17^{2}$ $17^{2}$ $17^{2}$ $17^{2}$ $34$ $17^{2}$ $34$ $34$ $17^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
17Data not computed