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Properties

Label 32.32.1569380774...0000.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{96}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $70.77$
Ramified primes $2, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1442401, 0, -25087872, 0, 159875504, 0, -516827632, 0, 986220191, 0, -1211673344, 0, 1012551072, 0, -596716576, 0, 253925217, 0, -79134848, 0, 18166288, 0, -3063248, 0, 374431, 0, -32256, 0, 1856, 0, -64, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 64*x^30 + 1856*x^28 - 32256*x^26 + 374431*x^24 - 3063248*x^22 + 18166288*x^20 - 79134848*x^18 + 253925217*x^16 - 596716576*x^14 + 1012551072*x^12 - 1211673344*x^10 + 986220191*x^8 - 516827632*x^6 + 159875504*x^4 - 25087872*x^2 + 1442401)
 
gp: K = bnfinit(x^32 - 64*x^30 + 1856*x^28 - 32256*x^26 + 374431*x^24 - 3063248*x^22 + 18166288*x^20 - 79134848*x^18 + 253925217*x^16 - 596716576*x^14 + 1012551072*x^12 - 1211673344*x^10 + 986220191*x^8 - 516827632*x^6 + 159875504*x^4 - 25087872*x^2 + 1442401, 1)
 

Normalized defining polynomial

\( x^{32} - 64 x^{30} + 1856 x^{28} - 32256 x^{26} + 374431 x^{24} - 3063248 x^{22} + 18166288 x^{20} - 79134848 x^{18} + 253925217 x^{16} - 596716576 x^{14} + 1012551072 x^{12} - 1211673344 x^{10} + 986220191 x^{8} - 516827632 x^{6} + 159875504 x^{4} - 25087872 x^{2} + 1442401 \)

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:  \(156938077449417789520626992646455296000000000000000000000000=2^{96}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.77$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(560=2^{4}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{560}(1,·)$, $\chi_{560}(517,·)$, $\chi_{560}(391,·)$, $\chi_{560}(139,·)$, $\chi_{560}(13,·)$, $\chi_{560}(531,·)$, $\chi_{560}(407,·)$, $\chi_{560}(153,·)$, $\chi_{560}(29,·)$, $\chi_{560}(419,·)$, $\chi_{560}(293,·)$, $\chi_{560}(169,·)$, $\chi_{560}(43,·)$, $\chi_{560}(559,·)$, $\chi_{560}(433,·)$, $\chi_{560}(309,·)$, $\chi_{560}(183,·)$, $\chi_{560}(279,·)$, $\chi_{560}(449,·)$, $\chi_{560}(323,·)$, $\chi_{560}(141,·)$, $\chi_{560}(547,·)$, $\chi_{560}(267,·)$, $\chi_{560}(463,·)$, $\chi_{560}(421,·)$, $\chi_{560}(97,·)$, $\chi_{560}(237,·)$, $\chi_{560}(111,·)$, $\chi_{560}(377,·)$, $\chi_{560}(251,·)$, $\chi_{560}(281,·)$, $\chi_{560}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{14} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{14} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{19} + \frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{627} a^{20} - \frac{40}{627} a^{18} + \frac{53}{627} a^{16} - \frac{130}{627} a^{14} + \frac{34}{627} a^{12} - \frac{20}{57} a^{10} - \frac{2}{19} a^{8} + \frac{1}{19} a^{6} - \frac{3}{19} a^{4} + \frac{214}{627} a^{2} + \frac{167}{627}$, $\frac{1}{753027} a^{21} - \frac{90119}{753027} a^{19} - \frac{30148}{251009} a^{17} - \frac{305479}{753027} a^{15} - \frac{170510}{753027} a^{13} + \frac{17707}{68457} a^{11} + \frac{3646}{22819} a^{9} + \frac{25007}{68457} a^{7} + \frac{31930}{68457} a^{5} - \frac{99134}{251009} a^{3} - \frac{232868}{753027} a$, $\frac{1}{753027} a^{22} - \frac{4}{68457} a^{20} + \frac{23897}{251009} a^{18} - \frac{2614}{39633} a^{16} + \frac{168172}{753027} a^{14} + \frac{245219}{753027} a^{12} - \frac{3560}{22819} a^{10} + \frac{32213}{68457} a^{8} + \frac{28327}{68457} a^{6} - \frac{59501}{251009} a^{4} + \frac{217507}{753027} a^{2} - \frac{5}{209}$, $\frac{1}{753027} a^{23} + \frac{122599}{753027} a^{19} - \frac{13058}{753027} a^{17} + \frac{10191}{251009} a^{15} - \frac{4017}{13211} a^{13} - \frac{7418}{68457} a^{11} - \frac{3784}{22819} a^{9} - \frac{4105}{22819} a^{7} + \frac{215077}{753027} a^{5} + \frac{61429}{251009} a^{3} + \frac{278171}{753027} a$, $\frac{1}{753027} a^{24} + \frac{97}{753027} a^{20} + \frac{117851}{753027} a^{18} + \frac{3379}{39633} a^{16} + \frac{133733}{753027} a^{14} + \frac{6829}{251009} a^{12} - \frac{155}{3603} a^{10} + \frac{15308}{68457} a^{8} - \frac{313363}{753027} a^{6} + \frac{87851}{251009} a^{4} + \frac{55893}{251009} a^{2} - \frac{35}{209}$, $\frac{1}{753027} a^{25} + \frac{24693}{251009} a^{19} + \frac{17318}{251009} a^{17} + \frac{146134}{753027} a^{15} + \frac{244372}{753027} a^{13} - \frac{31918}{68457} a^{11} + \frac{1332}{22819} a^{9} + \frac{121383}{251009} a^{7} + \frac{80458}{753027} a^{5} - \frac{101371}{753027} a^{3} - \frac{128719}{753027} a$, $\frac{1}{753027} a^{26} - \frac{383}{753027} a^{20} + \frac{1666}{68457} a^{18} - \frac{11739}{251009} a^{16} + \frac{45027}{251009} a^{14} + \frac{129302}{753027} a^{12} - \frac{12818}{68457} a^{10} + \frac{2484}{251009} a^{8} - \frac{117707}{753027} a^{6} - \frac{259903}{753027} a^{4} - \frac{83340}{251009} a^{2} + \frac{305}{627}$, $\frac{1}{753027} a^{27} - \frac{109018}{753027} a^{19} - \frac{12009}{251009} a^{17} + \frac{18833}{39633} a^{15} - \frac{54899}{251009} a^{13} - \frac{31099}{68457} a^{11} - \frac{347419}{753027} a^{9} - \frac{62332}{251009} a^{7} - \frac{279637}{753027} a^{5} + \frac{4919}{68457} a^{3} + \frac{95352}{251009} a$, $\frac{1}{753027} a^{28} + \frac{91}{251009} a^{20} + \frac{10045}{68457} a^{18} - \frac{124975}{753027} a^{16} - \frac{105341}{251009} a^{14} - \frac{140321}{753027} a^{12} + \frac{207443}{753027} a^{10} - \frac{371950}{753027} a^{8} + \frac{21283}{251009} a^{6} + \frac{25336}{68457} a^{4} + \frac{330493}{753027} a^{2} - \frac{20}{209}$, $\frac{1}{753027} a^{29} + \frac{114100}{753027} a^{19} - \frac{32645}{753027} a^{17} - \frac{385}{68457} a^{15} - \frac{9252}{251009} a^{13} + \frac{82410}{251009} a^{11} + \frac{8707}{39633} a^{9} - \frac{231490}{753027} a^{7} + \frac{2485}{68457} a^{5} - \frac{102565}{251009} a^{3} + \frac{82212}{251009} a$, $\frac{1}{753027} a^{30} + \frac{5}{753027} a^{20} + \frac{4331}{251009} a^{18} - \frac{9018}{251009} a^{16} - \frac{255946}{753027} a^{14} + \frac{133135}{753027} a^{12} - \frac{5905}{13211} a^{10} - \frac{231490}{753027} a^{8} + \frac{2485}{68457} a^{6} - \frac{102565}{251009} a^{4} - \frac{72830}{753027} a^{2} - \frac{10}{33}$, $\frac{1}{753027} a^{31} - \frac{12810}{251009} a^{19} - \frac{76852}{753027} a^{17} + \frac{267413}{753027} a^{15} - \frac{269360}{753027} a^{13} - \frac{18475}{251009} a^{11} + \frac{170956}{753027} a^{9} - \frac{1646}{3603} a^{7} + \frac{195236}{753027} a^{5} + \frac{53045}{251009} a^{3} + \frac{61041}{251009} a$

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:  $31$
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:  \( 23306007003525570000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

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 $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{35}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{70}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{2}, \sqrt{35})\), \(\Q(\zeta_{16})^+\), 4.4.2508800.1, \(\Q(\sqrt{10}, \sqrt{14})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{7})\), 4.4.51200.1, 4.4.100352.1, \(\Q(\sqrt{7}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{14})\), 4.4.256000.1, 4.4.12544000.2, 4.4.256000.2, 4.4.12544000.1, 4.4.8000.1, 4.4.392000.1, \(\Q(\zeta_{20})^+\), 4.4.6125.1, 8.8.25176309760000.2, 8.8.98344960000.1, 8.8.25176309760000.5, 8.8.2621440000.1, 8.8.40282095616.1, 8.8.6294077440000.1, 8.8.25176309760000.1, 8.8.629407744000000.2, 8.8.629407744000000.3, 8.8.2458624000000.1, 8.8.9604000000.1, 8.8.65536000000.1, 8.8.157351936000000.4, \(\Q(\zeta_{40})^+\), 8.8.153664000000.1, 8.8.629407744000000.4, 8.8.629407744000000.5, 8.8.153664000000.2, 8.8.2458624000000.2, 16.16.633846573131471257600000000.1, 16.16.396154108207169536000000000000.2, 16.16.6044831973376000000000000.1, \(\Q(\zeta_{80})^+\), 16.16.24759631762948096000000000000.2, 16.16.24759631762948096000000000000.1, 16.16.396154108207169536000000000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
2Data not computed
5Data not computed
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$