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Properties

Label 34.34.4556677242...8432.1
Degree $34$
Signature $[34, 0]$
Discriminant $2^{34}\cdot 103^{33}$
Root discriminant $179.75$
Ramified primes $2, 103$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![-103, 0, 58813, 0, -2421530, 0, 38356067, 0, -297008328, 0, 1228671035, 0, -2861712448, 0, 3877277925, 0, -3162397876, 0, 1616755259, 0, -536414833, 0, 118248223, 0, -17499597, 0, 1731739, 0, -112167, 0, 4532, 0, -103, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^34 - 103*x^32 + 4532*x^30 - 112167*x^28 + 1731739*x^26 - 17499597*x^24 + 118248223*x^22 - 536414833*x^20 + 1616755259*x^18 - 3162397876*x^16 + 3877277925*x^14 - 2861712448*x^12 + 1228671035*x^10 - 297008328*x^8 + 38356067*x^6 - 2421530*x^4 + 58813*x^2 - 103)
 
gp: K = bnfinit(x^34 - 103*x^32 + 4532*x^30 - 112167*x^28 + 1731739*x^26 - 17499597*x^24 + 118248223*x^22 - 536414833*x^20 + 1616755259*x^18 - 3162397876*x^16 + 3877277925*x^14 - 2861712448*x^12 + 1228671035*x^10 - 297008328*x^8 + 38356067*x^6 - 2421530*x^4 + 58813*x^2 - 103, 1)
 

Normalized defining polynomial

\( x^{34} - 103 x^{32} + 4532 x^{30} - 112167 x^{28} + 1731739 x^{26} - 17499597 x^{24} + 118248223 x^{22} - 536414833 x^{20} + 1616755259 x^{18} - 3162397876 x^{16} + 3877277925 x^{14} - 2861712448 x^{12} + 1228671035 x^{10} - 297008328 x^{8} + 38356067 x^{6} - 2421530 x^{4} + 58813 x^{2} - 103 \)

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

Invariants

Degree:  $34$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[34, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(45566772427063766319261485320198322575848903093077202070065748357513511698432=2^{34}\cdot 103^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $179.75$
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:  \(412=2^{2}\cdot 103\)
Dirichlet character group:    $\lbrace$$\chi_{412}(1,·)$, $\chi_{412}(3,·)$, $\chi_{412}(133,·)$, $\chi_{412}(385,·)$, $\chi_{412}(9,·)$, $\chi_{412}(13,·)$, $\chi_{412}(399,·)$, $\chi_{412}(275,·)$, $\chi_{412}(381,·)$, $\chi_{412}(279,·)$, $\chi_{412}(409,·)$, $\chi_{412}(27,·)$, $\chi_{412}(285,·)$, $\chi_{412}(31,·)$, $\chi_{412}(411,·)$, $\chi_{412}(295,·)$, $\chi_{412}(169,·)$, $\chi_{412}(137,·)$, $\chi_{412}(95,·)$, $\chi_{412}(61,·)$, $\chi_{412}(319,·)$, $\chi_{412}(373,·)$, $\chi_{412}(331,·)$, $\chi_{412}(403,·)$, $\chi_{412}(81,·)$, $\chi_{412}(93,·)$, $\chi_{412}(351,·)$, $\chi_{412}(229,·)$, $\chi_{412}(39,·)$, $\chi_{412}(317,·)$, $\chi_{412}(243,·)$, $\chi_{412}(117,·)$, $\chi_{412}(183,·)$, $\chi_{412}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{47} a^{24} + \frac{19}{47} a^{22} + \frac{14}{47} a^{20} - \frac{12}{47} a^{18} + \frac{6}{47} a^{16} + \frac{16}{47} a^{14} + \frac{9}{47} a^{12} + \frac{6}{47} a^{10} - \frac{4}{47} a^{8} - \frac{18}{47} a^{6} + \frac{10}{47} a^{4} - \frac{18}{47} a^{2} + \frac{18}{47}$, $\frac{1}{47} a^{25} + \frac{19}{47} a^{23} + \frac{14}{47} a^{21} - \frac{12}{47} a^{19} + \frac{6}{47} a^{17} + \frac{16}{47} a^{15} + \frac{9}{47} a^{13} + \frac{6}{47} a^{11} - \frac{4}{47} a^{9} - \frac{18}{47} a^{7} + \frac{10}{47} a^{5} - \frac{18}{47} a^{3} + \frac{18}{47} a$, $\frac{1}{47} a^{26} - \frac{18}{47} a^{22} + \frac{4}{47} a^{20} - \frac{1}{47} a^{18} - \frac{4}{47} a^{16} - \frac{13}{47} a^{14} + \frac{23}{47} a^{12} + \frac{23}{47} a^{10} + \frac{11}{47} a^{8} + \frac{23}{47} a^{6} - \frac{20}{47} a^{4} - \frac{16}{47} a^{2} - \frac{13}{47}$, $\frac{1}{47} a^{27} - \frac{18}{47} a^{23} + \frac{4}{47} a^{21} - \frac{1}{47} a^{19} - \frac{4}{47} a^{17} - \frac{13}{47} a^{15} + \frac{23}{47} a^{13} + \frac{23}{47} a^{11} + \frac{11}{47} a^{9} + \frac{23}{47} a^{7} - \frac{20}{47} a^{5} - \frac{16}{47} a^{3} - \frac{13}{47} a$, $\frac{1}{47} a^{28} + \frac{17}{47} a^{22} + \frac{16}{47} a^{20} + \frac{15}{47} a^{18} + \frac{1}{47} a^{16} - \frac{18}{47} a^{14} - \frac{3}{47} a^{12} - \frac{22}{47} a^{10} - \frac{2}{47} a^{8} - \frac{15}{47} a^{6} + \frac{23}{47} a^{4} - \frac{8}{47} a^{2} - \frac{5}{47}$, $\frac{1}{47} a^{29} + \frac{17}{47} a^{23} + \frac{16}{47} a^{21} + \frac{15}{47} a^{19} + \frac{1}{47} a^{17} - \frac{18}{47} a^{15} - \frac{3}{47} a^{13} - \frac{22}{47} a^{11} - \frac{2}{47} a^{9} - \frac{15}{47} a^{7} + \frac{23}{47} a^{5} - \frac{8}{47} a^{3} - \frac{5}{47} a$, $\frac{1}{47} a^{30} + \frac{22}{47} a^{22} + \frac{12}{47} a^{20} + \frac{17}{47} a^{18} + \frac{21}{47} a^{16} + \frac{7}{47} a^{14} + \frac{13}{47} a^{12} - \frac{10}{47} a^{10} + \frac{6}{47} a^{8} + \frac{10}{47} a^{4} + \frac{19}{47} a^{2} + \frac{23}{47}$, $\frac{1}{47} a^{31} + \frac{22}{47} a^{23} + \frac{12}{47} a^{21} + \frac{17}{47} a^{19} + \frac{21}{47} a^{17} + \frac{7}{47} a^{15} + \frac{13}{47} a^{13} - \frac{10}{47} a^{11} + \frac{6}{47} a^{9} + \frac{10}{47} a^{5} + \frac{19}{47} a^{3} + \frac{23}{47} a$, $\frac{1}{9437887215720177534459648901415421112502535037122562836599} a^{32} + \frac{55557609152963376715994584539988286080305750162613882741}{9437887215720177534459648901415421112502535037122562836599} a^{30} + \frac{92465966130435395876201684673004932886818776589730127721}{9437887215720177534459648901415421112502535037122562836599} a^{28} - \frac{42027633211984960628885734783475497369981750302742683709}{9437887215720177534459648901415421112502535037122562836599} a^{26} + \frac{27033156525732454198857302653179778743101239814602333897}{9437887215720177534459648901415421112502535037122562836599} a^{24} - \frac{809696357795395874918665426289096955879063000515747515050}{9437887215720177534459648901415421112502535037122562836599} a^{22} + \frac{2452205755281258519013504577264205000924746922863923285559}{9437887215720177534459648901415421112502535037122562836599} a^{20} - \frac{2430934062068386546112299107781361617047079566775499105968}{9437887215720177534459648901415421112502535037122562836599} a^{18} + \frac{898659060638724774320703912112700302385502267068082584018}{9437887215720177534459648901415421112502535037122562836599} a^{16} + \frac{1026416290167964887641791808437038869444877446632973480194}{9437887215720177534459648901415421112502535037122562836599} a^{14} - \frac{3738260839932229132739609240694417962620415629098520585161}{9437887215720177534459648901415421112502535037122562836599} a^{12} + \frac{16048353966610410560577089531767510251250909555674175029}{35885502721369496328743912172682209553241578087918489873} a^{10} - \frac{1376039481348883755056360377207481941370859779010147022331}{9437887215720177534459648901415421112502535037122562836599} a^{8} - \frac{2927934930319072559238080849209544224294742631898977463047}{9437887215720177534459648901415421112502535037122562836599} a^{6} - \frac{1946983127026010157120110086019470776782996585224698433298}{9437887215720177534459648901415421112502535037122562836599} a^{4} - \frac{2380576461860352243844087620305736245720310018875908070}{11467663688602888863255952492606830027342083884717573313} a^{2} - \frac{2353285574092495377539882830921654824867879186723976348837}{9437887215720177534459648901415421112502535037122562836599}$, $\frac{1}{9437887215720177534459648901415421112502535037122562836599} a^{33} + \frac{55557609152963376715994584539988286080305750162613882741}{9437887215720177534459648901415421112502535037122562836599} a^{31} + \frac{92465966130435395876201684673004932886818776589730127721}{9437887215720177534459648901415421112502535037122562836599} a^{29} - \frac{42027633211984960628885734783475497369981750302742683709}{9437887215720177534459648901415421112502535037122562836599} a^{27} + \frac{27033156525732454198857302653179778743101239814602333897}{9437887215720177534459648901415421112502535037122562836599} a^{25} - \frac{809696357795395874918665426289096955879063000515747515050}{9437887215720177534459648901415421112502535037122562836599} a^{23} + \frac{2452205755281258519013504577264205000924746922863923285559}{9437887215720177534459648901415421112502535037122562836599} a^{21} - \frac{2430934062068386546112299107781361617047079566775499105968}{9437887215720177534459648901415421112502535037122562836599} a^{19} + \frac{898659060638724774320703912112700302385502267068082584018}{9437887215720177534459648901415421112502535037122562836599} a^{17} + \frac{1026416290167964887641791808437038869444877446632973480194}{9437887215720177534459648901415421112502535037122562836599} a^{15} - \frac{3738260839932229132739609240694417962620415629098520585161}{9437887215720177534459648901415421112502535037122562836599} a^{13} + \frac{16048353966610410560577089531767510251250909555674175029}{35885502721369496328743912172682209553241578087918489873} a^{11} - \frac{1376039481348883755056360377207481941370859779010147022331}{9437887215720177534459648901415421112502535037122562836599} a^{9} - \frac{2927934930319072559238080849209544224294742631898977463047}{9437887215720177534459648901415421112502535037122562836599} a^{7} - \frac{1946983127026010157120110086019470776782996585224698433298}{9437887215720177534459648901415421112502535037122562836599} a^{5} - \frac{2380576461860352243844087620305736245720310018875908070}{11467663688602888863255952492606830027342083884717573313} a^{3} - \frac{2353285574092495377539882830921654824867879186723976348837}{9437887215720177534459648901415421112502535037122562836599} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  $33$
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:  \( 6234441083346011000000000000 \) (assuming GRH)
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

\(\Q(\sqrt{103}) \), 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 $17^{2}$ $34$ $34$ $17^{2}$ $17^{2}$ $17^{2}$ $34$ $34$ $17^{2}$ $17^{2}$ $34$ $17^{2}$ $17^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{34}$ $34$ $34$

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