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Properties

Label 34.0.59725279955...9104.1
Degree $34$
Signature $[0, 17]$
Discriminant $-\,2^{51}\cdot 103^{33}$
Root discriminant $254.20$
Ramified primes $2, 103$
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![13500416, 0, 3854368768, 0, 79348695040, 0, 628425801728, 0, 2433092222976, 0, 5032636559360, 0, 5860787093504, 0, 3970332595200, 0, 1619147712512, 0, 413889346304, 0, 68661098624, 0, 7567886272, 0, 559987104, 0, 27707824, 0, 897336, 0, 18128, 0, 206, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^34 + 206*x^32 + 18128*x^30 + 897336*x^28 + 27707824*x^26 + 559987104*x^24 + 7567886272*x^22 + 68661098624*x^20 + 413889346304*x^18 + 1619147712512*x^16 + 3970332595200*x^14 + 5860787093504*x^12 + 5032636559360*x^10 + 2433092222976*x^8 + 628425801728*x^6 + 79348695040*x^4 + 3854368768*x^2 + 13500416)
 
gp: K = bnfinit(x^34 + 206*x^32 + 18128*x^30 + 897336*x^28 + 27707824*x^26 + 559987104*x^24 + 7567886272*x^22 + 68661098624*x^20 + 413889346304*x^18 + 1619147712512*x^16 + 3970332595200*x^14 + 5860787093504*x^12 + 5032636559360*x^10 + 2433092222976*x^8 + 628425801728*x^6 + 79348695040*x^4 + 3854368768*x^2 + 13500416, 1)
 

Normalized defining polynomial

\( x^{34} + 206 x^{32} + 18128 x^{30} + 897336 x^{28} + 27707824 x^{26} + 559987104 x^{24} + 7567886272 x^{22} + 68661098624 x^{20} + 413889346304 x^{18} + 1619147712512 x^{16} + 3970332595200 x^{14} + 5860787093504 x^{12} + 5032636559360 x^{10} + 2433092222976 x^{8} + 628425801728 x^{6} + 79348695040 x^{4} + 3854368768 x^{2} + 13500416 \)

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:  \(-5972527995560101978998241403889034536661667426215815029727657768716011005336879104=-\,2^{51}\cdot 103^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $254.20$
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:  $2, 103$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(824=2^{3}\cdot 103\)
Dirichlet character group:    $\lbrace$$\chi_{824}(1,·)$, $\chi_{824}(473,·)$, $\chi_{824}(645,·)$, $\chi_{824}(641,·)$, $\chi_{824}(9,·)$, $\chi_{824}(525,·)$, $\chi_{824}(301,·)$, $\chi_{824}(529,·)$, $\chi_{824}(793,·)$, $\chi_{824}(409,·)$, $\chi_{824}(69,·)$, $\chi_{824}(389,·)$, $\chi_{824}(545,·)$, $\chi_{824}(37,·)$, $\chi_{824}(169,·)$, $\chi_{824}(557,·)$, $\chi_{824}(385,·)$, $\chi_{824}(137,·)$, $\chi_{824}(697,·)$, $\chi_{824}(245,·)$, $\chi_{824}(197,·)$, $\chi_{824}(333,·)$, $\chi_{824}(205,·)$, $\chi_{824}(81,·)$, $\chi_{824}(729,·)$, $\chi_{824}(605,·)$, $\chi_{824}(621,·)$, $\chi_{824}(485,·)$, $\chi_{824}(785,·)$, $\chi_{824}(237,·)$, $\chi_{824}(501,·)$, $\chi_{824}(425,·)$, $\chi_{824}(505,·)$, $\chi_{824}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{192512} a^{24} - \frac{19}{96256} a^{22} + \frac{7}{24064} a^{20} + \frac{3}{6016} a^{18} + \frac{3}{6016} a^{16} - \frac{1}{376} a^{14} + \frac{9}{3008} a^{12} - \frac{3}{752} a^{10} - \frac{1}{188} a^{8} + \frac{9}{188} a^{6} + \frac{5}{94} a^{4} + \frac{9}{47} a^{2} + \frac{18}{47}$, $\frac{1}{192512} a^{25} - \frac{19}{96256} a^{23} + \frac{7}{24064} a^{21} + \frac{3}{6016} a^{19} + \frac{3}{6016} a^{17} - \frac{1}{376} a^{15} + \frac{9}{3008} a^{13} - \frac{3}{752} a^{11} - \frac{1}{188} a^{9} + \frac{9}{188} a^{7} + \frac{5}{94} a^{5} + \frac{9}{47} a^{3} + \frac{18}{47} a$, $\frac{1}{385024} a^{26} - \frac{9}{48128} a^{22} - \frac{1}{12032} a^{20} - \frac{1}{24064} a^{18} + \frac{1}{3008} a^{16} - \frac{13}{6016} a^{14} - \frac{23}{3008} a^{12} + \frac{23}{1504} a^{10} - \frac{11}{752} a^{8} + \frac{23}{376} a^{6} + \frac{5}{47} a^{4} - \frac{8}{47} a^{2} + \frac{13}{47}$, $\frac{1}{385024} a^{27} - \frac{9}{48128} a^{23} - \frac{1}{12032} a^{21} - \frac{1}{24064} a^{19} + \frac{1}{3008} a^{17} - \frac{13}{6016} a^{15} - \frac{23}{3008} a^{13} + \frac{23}{1504} a^{11} - \frac{11}{752} a^{9} + \frac{23}{376} a^{7} + \frac{5}{47} a^{5} - \frac{8}{47} a^{3} + \frac{13}{47} a$, $\frac{1}{770048} a^{28} - \frac{17}{96256} a^{22} + \frac{1}{3008} a^{20} - \frac{15}{24064} a^{18} + \frac{1}{12032} a^{16} + \frac{9}{3008} a^{14} - \frac{3}{3008} a^{12} + \frac{11}{752} a^{10} - \frac{1}{376} a^{8} + \frac{15}{376} a^{6} + \frac{23}{188} a^{4} + \frac{4}{47} a^{2} - \frac{5}{47}$, $\frac{1}{770048} a^{29} - \frac{17}{96256} a^{23} + \frac{1}{3008} a^{21} - \frac{15}{24064} a^{19} + \frac{1}{12032} a^{17} + \frac{9}{3008} a^{15} - \frac{3}{3008} a^{13} + \frac{11}{752} a^{11} - \frac{1}{376} a^{9} + \frac{15}{376} a^{7} + \frac{23}{188} a^{5} + \frac{4}{47} a^{3} - \frac{5}{47} a$, $\frac{1}{1540096} a^{30} + \frac{11}{48128} a^{22} - \frac{3}{12032} a^{20} + \frac{17}{24064} a^{18} - \frac{21}{12032} a^{16} + \frac{7}{6016} a^{14} - \frac{13}{3008} a^{12} - \frac{5}{752} a^{10} - \frac{3}{376} a^{8} - \frac{5}{94} a^{4} + \frac{19}{94} a^{2} - \frac{23}{47}$, $\frac{1}{1540096} a^{31} + \frac{11}{48128} a^{23} - \frac{3}{12032} a^{21} + \frac{17}{24064} a^{19} - \frac{21}{12032} a^{17} + \frac{7}{6016} a^{15} - \frac{13}{3008} a^{13} - \frac{5}{752} a^{11} - \frac{3}{376} a^{9} - \frac{5}{94} a^{5} + \frac{19}{94} a^{3} - \frac{23}{47} a$, $\frac{1}{618521376569437554898347550403161038028966136192864278059352064} a^{32} - \frac{55557609152963376715994584539988286080305750162613882741}{309260688284718777449173775201580519014483068096432139029676032} a^{30} + \frac{92465966130435395876201684673004932886818776589730127721}{154630344142359388724586887600790259507241534048216069514838016} a^{28} + \frac{42027633211984960628885734783475497369981750302742683709}{77315172071179694362293443800395129753620767024108034757419008} a^{26} + \frac{27033156525732454198857302653179778743101239814602333897}{38657586035589847181146721900197564876810383512054017378709504} a^{24} + \frac{404848178897697937459332713144548477939531500257873757525}{9664396508897461795286680475049391219202595878013504344677376} a^{22} + \frac{2452205755281258519013504577264205000924746922863923285559}{9664396508897461795286680475049391219202595878013504344677376} a^{20} + \frac{151933378879274159132018694236335101065442472923468694123}{302012390903045681102708764845293475600081121187922010771168} a^{18} + \frac{449329530319362387160351956056350151192751133534041292009}{1208049563612182724410835059381173902400324484751688043084672} a^{16} - \frac{513208145083982443820895904218519434722438723316486740097}{604024781806091362205417529690586951200162242375844021542336} a^{14} - \frac{3738260839932229132739609240694417962620415629098520585161}{604024781806091362205417529690586951200162242375844021542336} a^{12} - \frac{16048353966610410560577089531767510251250909555674175029}{1148336087083823882519805189525830705703730498813391675936} a^{10} - \frac{1376039481348883755056360377207481941370859779010147022331}{151006195451522840551354382422646737800040560593961005385584} a^{8} + \frac{2927934930319072559238080849209544224294742631898977463047}{75503097725761420275677191211323368900020280296980502692792} a^{6} - \frac{973491563513005078560055043009735388391498292612349216649}{18875774431440355068919297802830842225005070074245125673198} a^{4} + \frac{1190288230930176121922043810152868122860155009437954035}{11467663688602888863255952492606830027342083884717573313} a^{2} - \frac{2353285574092495377539882830921654824867879186723976348837}{9437887215720177534459648901415421112502535037122562836599}$, $\frac{1}{618521376569437554898347550403161038028966136192864278059352064} a^{33} - \frac{55557609152963376715994584539988286080305750162613882741}{309260688284718777449173775201580519014483068096432139029676032} a^{31} + \frac{92465966130435395876201684673004932886818776589730127721}{154630344142359388724586887600790259507241534048216069514838016} a^{29} + \frac{42027633211984960628885734783475497369981750302742683709}{77315172071179694362293443800395129753620767024108034757419008} a^{27} + \frac{27033156525732454198857302653179778743101239814602333897}{38657586035589847181146721900197564876810383512054017378709504} a^{25} + \frac{404848178897697937459332713144548477939531500257873757525}{9664396508897461795286680475049391219202595878013504344677376} a^{23} + \frac{2452205755281258519013504577264205000924746922863923285559}{9664396508897461795286680475049391219202595878013504344677376} a^{21} + \frac{151933378879274159132018694236335101065442472923468694123}{302012390903045681102708764845293475600081121187922010771168} a^{19} + \frac{449329530319362387160351956056350151192751133534041292009}{1208049563612182724410835059381173902400324484751688043084672} a^{17} - \frac{513208145083982443820895904218519434722438723316486740097}{604024781806091362205417529690586951200162242375844021542336} a^{15} - \frac{3738260839932229132739609240694417962620415629098520585161}{604024781806091362205417529690586951200162242375844021542336} a^{13} - \frac{16048353966610410560577089531767510251250909555674175029}{1148336087083823882519805189525830705703730498813391675936} a^{11} - \frac{1376039481348883755056360377207481941370859779010147022331}{151006195451522840551354382422646737800040560593961005385584} a^{9} + \frac{2927934930319072559238080849209544224294742631898977463047}{75503097725761420275677191211323368900020280296980502692792} a^{7} - \frac{973491563513005078560055043009735388391498292612349216649}{18875774431440355068919297802830842225005070074245125673198} a^{5} + \frac{1190288230930176121922043810152868122860155009437954035}{11467663688602888863255952492606830027342083884717573313} a^{3} - \frac{2353285574092495377539882830921654824867879186723976348837}{9437887215720177534459648901415421112502535037122562836599} 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

\(\Q(\sqrt{-206}) \), 17.17.160470643909878751793805444097921.1

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 $17^{2}$ $17^{2}$ $17^{2}$ $17^{2}$ $34$ $17^{2}$ $34$ $17^{2}$ $34$ $34$ $17^{2}$ $17^{2}$ $17^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{17}$ $17^{2}$ $34$

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
103Data not computed