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Properties

Label 32.32.1172131553...2529.1
Degree $32$
Signature $[32, 0]$
Discriminant $3^{16}\cdot 7^{16}\cdot 17^{30}$
Root discriminant $65.26$
Ramified primes $3, 7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_{16}$ (as 32T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![76603, 103231, -3752014, -1257219, 44453923, 9562400, -205111641, -39835109, 490797914, 90494753, -702418817, -122456240, 653702339, 106258835, -417033506, -62155621, 188675045, 25303905, -61804530, -7300125, 14806525, 1501280, -2594055, -218555, 328550, 21999, -29279, -1456, 1741, 57, -62, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 - 62*x^30 + 57*x^29 + 1741*x^28 - 1456*x^27 - 29279*x^26 + 21999*x^25 + 328550*x^24 - 218555*x^23 - 2594055*x^22 + 1501280*x^21 + 14806525*x^20 - 7300125*x^19 - 61804530*x^18 + 25303905*x^17 + 188675045*x^16 - 62155621*x^15 - 417033506*x^14 + 106258835*x^13 + 653702339*x^12 - 122456240*x^11 - 702418817*x^10 + 90494753*x^9 + 490797914*x^8 - 39835109*x^7 - 205111641*x^6 + 9562400*x^5 + 44453923*x^4 - 1257219*x^3 - 3752014*x^2 + 103231*x + 76603)
 
gp: K = bnfinit(x^32 - x^31 - 62*x^30 + 57*x^29 + 1741*x^28 - 1456*x^27 - 29279*x^26 + 21999*x^25 + 328550*x^24 - 218555*x^23 - 2594055*x^22 + 1501280*x^21 + 14806525*x^20 - 7300125*x^19 - 61804530*x^18 + 25303905*x^17 + 188675045*x^16 - 62155621*x^15 - 417033506*x^14 + 106258835*x^13 + 653702339*x^12 - 122456240*x^11 - 702418817*x^10 + 90494753*x^9 + 490797914*x^8 - 39835109*x^7 - 205111641*x^6 + 9562400*x^5 + 44453923*x^4 - 1257219*x^3 - 3752014*x^2 + 103231*x + 76603, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} - 62 x^{30} + 57 x^{29} + 1741 x^{28} - 1456 x^{27} - 29279 x^{26} + 21999 x^{25} + 328550 x^{24} - 218555 x^{23} - 2594055 x^{22} + 1501280 x^{21} + 14806525 x^{20} - 7300125 x^{19} - 61804530 x^{18} + 25303905 x^{17} + 188675045 x^{16} - 62155621 x^{15} - 417033506 x^{14} + 106258835 x^{13} + 653702339 x^{12} - 122456240 x^{11} - 702418817 x^{10} + 90494753 x^{9} + 490797914 x^{8} - 39835109 x^{7} - 205111641 x^{6} + 9562400 x^{5} + 44453923 x^{4} - 1257219 x^{3} - 3752014 x^{2} + 103231 x + 76603 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[32, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11721315531921426838644724774450299033325694662265349722529=3^{16}\cdot 7^{16}\cdot 17^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.26$
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:  $3, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(357=3\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{357}(1,·)$, $\chi_{357}(260,·)$, $\chi_{357}(265,·)$, $\chi_{357}(139,·)$, $\chi_{357}(274,·)$, $\chi_{357}(29,·)$, $\chi_{357}(286,·)$, $\chi_{357}(160,·)$, $\chi_{357}(293,·)$, $\chi_{357}(169,·)$, $\chi_{357}(43,·)$, $\chi_{357}(176,·)$, $\chi_{357}(181,·)$, $\chi_{357}(314,·)$, $\chi_{357}(188,·)$, $\chi_{357}(64,·)$, $\chi_{357}(197,·)$, $\chi_{357}(71,·)$, $\chi_{357}(328,·)$, $\chi_{357}(83,·)$, $\chi_{357}(218,·)$, $\chi_{357}(92,·)$, $\chi_{357}(97,·)$, $\chi_{357}(356,·)$, $\chi_{357}(230,·)$, $\chi_{357}(104,·)$, $\chi_{357}(106,·)$, $\chi_{357}(113,·)$, $\chi_{357}(244,·)$, $\chi_{357}(251,·)$, $\chi_{357}(253,·)$, $\chi_{357}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{271} a^{17} - \frac{34}{271} a^{15} - \frac{66}{271} a^{13} - \frac{13}{271} a^{11} + \frac{55}{271} a^{9} - \frac{132}{271} a^{7} - \frac{103}{271} a^{5} - \frac{96}{271} a^{3} + \frac{16}{271} a + \frac{85}{271}$, $\frac{1}{271} a^{18} - \frac{34}{271} a^{16} - \frac{66}{271} a^{14} - \frac{13}{271} a^{12} + \frac{55}{271} a^{10} - \frac{132}{271} a^{8} - \frac{103}{271} a^{6} - \frac{96}{271} a^{4} + \frac{16}{271} a^{2} + \frac{85}{271} a$, $\frac{1}{271} a^{19} + \frac{133}{271} a^{15} - \frac{89}{271} a^{13} - \frac{116}{271} a^{11} + \frac{112}{271} a^{9} + \frac{16}{271} a^{7} - \frac{75}{271} a^{5} + \frac{4}{271} a^{3} + \frac{85}{271} a^{2} + \frac{2}{271} a - \frac{91}{271}$, $\frac{1}{271} a^{20} + \frac{133}{271} a^{16} - \frac{89}{271} a^{14} - \frac{116}{271} a^{12} + \frac{112}{271} a^{10} + \frac{16}{271} a^{8} - \frac{75}{271} a^{6} + \frac{4}{271} a^{4} + \frac{85}{271} a^{3} + \frac{2}{271} a^{2} - \frac{91}{271} a$, $\frac{1}{271} a^{21} + \frac{97}{271} a^{15} - \frac{10}{271} a^{13} - \frac{56}{271} a^{11} + \frac{18}{271} a^{9} - \frac{134}{271} a^{7} - \frac{118}{271} a^{5} + \frac{85}{271} a^{4} + \frac{33}{271} a^{3} - \frac{91}{271} a^{2} + \frac{40}{271} a + \frac{77}{271}$, $\frac{1}{271} a^{22} + \frac{97}{271} a^{16} - \frac{10}{271} a^{14} - \frac{56}{271} a^{12} + \frac{18}{271} a^{10} - \frac{134}{271} a^{8} - \frac{118}{271} a^{6} + \frac{85}{271} a^{5} + \frac{33}{271} a^{4} - \frac{91}{271} a^{3} + \frac{40}{271} a^{2} + \frac{77}{271} a$, $\frac{1}{271} a^{23} + \frac{36}{271} a^{15} + \frac{113}{271} a^{13} - \frac{76}{271} a^{11} - \frac{49}{271} a^{9} - \frac{51}{271} a^{7} + \frac{85}{271} a^{6} - \frac{3}{271} a^{5} - \frac{91}{271} a^{4} - \frac{133}{271} a^{3} + \frac{77}{271} a^{2} + \frac{74}{271} a - \frac{115}{271}$, $\frac{1}{271} a^{24} + \frac{36}{271} a^{16} + \frac{113}{271} a^{14} - \frac{76}{271} a^{12} - \frac{49}{271} a^{10} - \frac{51}{271} a^{8} + \frac{85}{271} a^{7} - \frac{3}{271} a^{6} - \frac{91}{271} a^{5} - \frac{133}{271} a^{4} + \frac{77}{271} a^{3} + \frac{74}{271} a^{2} - \frac{115}{271} a$, $\frac{1}{32756041} a^{25} - \frac{28718}{32756041} a^{24} + \frac{59357}{32756041} a^{23} - \frac{21303}{32756041} a^{22} - \frac{6306}{32756041} a^{21} - \frac{12577}{32756041} a^{20} - \frac{3378}{32756041} a^{19} + \frac{32559}{32756041} a^{18} + \frac{60405}{32756041} a^{17} + \frac{2940251}{32756041} a^{16} + \frac{8114700}{32756041} a^{15} + \frac{3402779}{32756041} a^{14} - \frac{33139}{32756041} a^{13} + \frac{11487011}{32756041} a^{12} + \frac{9646696}{32756041} a^{11} - \frac{7608424}{32756041} a^{10} + \frac{15413519}{32756041} a^{9} + \frac{2514567}{32756041} a^{8} + \frac{9528583}{32756041} a^{7} + \frac{10212859}{32756041} a^{6} + \frac{530337}{32756041} a^{5} + \frac{320113}{32756041} a^{4} + \frac{13747324}{32756041} a^{3} - \frac{8360037}{32756041} a^{2} - \frac{1626019}{32756041} a - \frac{12172209}{32756041}$, $\frac{1}{32756041} a^{26} + \frac{38666}{32756041} a^{24} - \frac{50690}{32756041} a^{23} - \frac{57729}{32756041} a^{22} - \frac{43527}{32756041} a^{21} - \frac{27116}{32756041} a^{20} - \frac{38303}{32756041} a^{19} + \frac{31711}{32756041} a^{18} + \frac{9545}{32756041} a^{17} - \frac{2698652}{32756041} a^{16} + \frac{13431253}{32756041} a^{15} + \frac{9945622}{32756041} a^{14} + \frac{6946365}{32756041} a^{13} + \frac{1858746}{32756041} a^{12} + \frac{4017211}{32756041} a^{11} + \frac{10364947}{32756041} a^{10} - \frac{7273443}{32756041} a^{9} - \frac{4811071}{32756041} a^{8} + \frac{11356456}{32756041} a^{7} - \frac{15658760}{32756041} a^{6} + \frac{2967757}{32756041} a^{5} - \frac{4101226}{32756041} a^{4} - \frac{360895}{32756041} a^{3} + \frac{11293621}{32756041} a^{2} + \frac{5554745}{32756041} a + \frac{8191457}{32756041}$, $\frac{1}{32756041} a^{27} + \frac{38492}{32756041} a^{24} - \frac{56943}{32756041} a^{23} + \frac{43277}{32756041} a^{22} + \frac{3873}{32756041} a^{21} - \frac{54}{32756041} a^{20} - \frac{16092}{32756041} a^{19} - \frac{45284}{32756041} a^{18} + \frac{51984}{32756041} a^{17} + \frac{11872105}{32756041} a^{16} + \frac{8490320}{32756041} a^{15} + \frac{10429311}{32756041} a^{14} - \frac{14701478}{32756041} a^{13} + \frac{6595261}{32756041} a^{12} - \frac{14478727}{32756041} a^{11} + \frac{12790982}{32756041} a^{10} - \frac{9890123}{32756041} a^{9} - \frac{13158192}{32756041} a^{8} - \frac{15635614}{32756041} a^{7} - \frac{5990951}{32756041} a^{6} - \frac{15345779}{32756041} a^{5} + \frac{3933345}{32756041} a^{4} + \frac{10089214}{32756041} a^{3} + \frac{364762}{32756041} a^{2} - \frac{9943544}{32756041} a - \frac{14634643}{32756041}$, $\frac{1}{32756041} a^{28} - \frac{8982}{32756041} a^{24} - \frac{22725}{32756041} a^{23} + \frac{10085}{32756041} a^{22} + \frac{21530}{32756041} a^{21} + \frac{9437}{32756041} a^{20} + \frac{44367}{32756041} a^{19} - \frac{18516}{32756041} a^{18} - \frac{7957}{32756041} a^{17} - \frac{8341843}{32756041} a^{16} + \frac{7103722}{32756041} a^{15} + \frac{8419958}{32756041} a^{14} + \frac{465565}{32756041} a^{13} - \frac{16114104}{32756041} a^{12} - \frac{14964069}{32756041} a^{11} + \frac{12263396}{32756041} a^{10} - \frac{8742748}{32756041} a^{9} - \frac{15148327}{32756041} a^{8} - \frac{9970613}{32756041} a^{7} - \frac{3280909}{32756041} a^{6} - \frac{11004144}{32756041} a^{5} + \frac{9405874}{32756041} a^{4} + \frac{9673418}{32756041} a^{3} - \frac{15218351}{32756041} a^{2} + \frac{10171266}{32756041} a - \frac{10104249}{32756041}$, $\frac{1}{32756041} a^{29} - \frac{29087}{32756041} a^{24} - \frac{7322}{32756041} a^{23} + \frac{16777}{32756041} a^{22} + \frac{57444}{32756041} a^{21} - \frac{28733}{32756041} a^{20} - \frac{21091}{32756041} a^{19} + \frac{50032}{32756041} a^{18} - \frac{33953}{32756041} a^{17} + \frac{13981319}{32756041} a^{16} + \frac{377033}{32756041} a^{15} - \frac{3686744}{32756041} a^{14} - \frac{4338842}{32756041} a^{13} + \frac{1717234}{32756041} a^{12} + \frac{9609614}{32756041} a^{11} - \frac{6781103}{32756041} a^{10} + \frac{15104133}{32756041} a^{9} + \frac{9012739}{32756041} a^{8} + \frac{102660}{32756041} a^{7} - \frac{12820475}{32756041} a^{6} - \frac{9247436}{32756041} a^{5} - \frac{15975616}{32756041} a^{4} + \frac{9353418}{32756041} a^{3} + \frac{9372875}{32756041} a^{2} + \frac{10263222}{32756041} a - \frac{9005288}{32756041}$, $\frac{1}{32756041} a^{30} + \frac{11693}{32756041} a^{24} + \frac{12472}{32756041} a^{23} + \frac{1829}{32756041} a^{22} + \frac{30823}{32756041} a^{21} + \frac{28227}{32756041} a^{20} - \frac{58602}{32756041} a^{19} - \frac{14605}{32756041} a^{18} - \frac{20338}{32756041} a^{17} + \frac{10488887}{32756041} a^{16} + \frac{8383554}{32756041} a^{15} + \frac{3360873}{32756041} a^{14} + \frac{8639013}{32756041} a^{13} + \frac{13186369}{32756041} a^{12} + \frac{13285376}{32756041} a^{11} + \frac{952497}{32756041} a^{10} - \frac{7056957}{32756041} a^{9} - \frac{9332727}{32756041} a^{8} - \frac{7053172}{32756041} a^{7} + \frac{1909246}{32756041} a^{6} - \frac{5467153}{32756041} a^{5} - \frac{7497434}{32756041} a^{4} - \frac{347592}{32756041} a^{3} - \frac{13425042}{32756041} a^{2} + \frac{7636460}{32756041} a - \frac{10038825}{32756041}$, $\frac{1}{32756041} a^{31} + \frac{32408}{32756041} a^{24} - \frac{18290}{32756041} a^{23} + \frac{11671}{32756041} a^{22} + \frac{32975}{32756041} a^{21} + \frac{25123}{32756041} a^{20} - \frac{40468}{32756041} a^{19} + \frac{10925}{32756041} a^{18} + \frac{27569}{32756041} a^{17} - \frac{1698184}{32756041} a^{16} + \frac{1667031}{32756041} a^{15} + \frac{14907851}{32756041} a^{14} + \frac{10629979}{32756041} a^{13} + \frac{7037469}{32756041} a^{12} - \frac{684018}{32756041} a^{11} - \frac{2327641}{32756041} a^{10} + \frac{928257}{32756041} a^{9} - \frac{2212545}{32756041} a^{8} - \frac{4538846}{32756041} a^{7} - \frac{13669991}{32756041} a^{6} + \frac{10084975}{32756041} a^{5} - \frac{4888871}{32756041} a^{4} + \frac{204588}{32756041} a^{3} - \frac{15595897}{32756041} a^{2} + \frac{3288852}{32756041} a - \frac{10346312}{32756041}$

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:  $31$
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:  \( 7704904687609626000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{16}$ (as 32T32):

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 $C_2\times C_{16}$
Character table for $C_2\times C_{16}$ is not computed

Intermediate fields

\(\Q(\sqrt{357}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{17}, \sqrt{21})\), 4.4.4913.1, 4.4.2166633.1, 8.8.4694298556689.1, \(\Q(\zeta_{17})^+\), 8.8.79803075463713.1, 16.16.6368530853467071849979746369.1, \(\Q(\zeta_{51})^+\), 16.16.16501299269766837593302193.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{4}$ R $16^{2}$ R $16^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
7Data not computed
17Data not computed