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Properties

Label 32.0.50522262278...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{64}\cdot 5^{24}\cdot 11^{16}$
Root discriminant $44.36$
Ramified primes $2, 5, 11$
Class number Not computed
Class group Not computed
Galois group $C_2^3\times C_4$ (as 32T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43046721, 0, 0, 0, -3720087, 0, 0, 0, -209952, 0, 0, 0, 64071, 0, 0, 0, -2945, 0, 0, 0, 791, 0, 0, 0, -32, 0, 0, 0, -7, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 7*x^28 - 32*x^24 + 791*x^20 - 2945*x^16 + 64071*x^12 - 209952*x^8 - 3720087*x^4 + 43046721)
 
gp: K = bnfinit(x^32 - 7*x^28 - 32*x^24 + 791*x^20 - 2945*x^16 + 64071*x^12 - 209952*x^8 - 3720087*x^4 + 43046721, 1)
 

Normalized defining polynomial

\( x^{32} - 7 x^{28} - 32 x^{24} + 791 x^{20} - 2945 x^{16} + 64071 x^{12} - 209952 x^{8} - 3720087 x^{4} + 43046721 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(50522262278163705147147943936000000000000000000000000=2^{64}\cdot 5^{24}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.36$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(440=2^{3}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{440}(1,·)$, $\chi_{440}(131,·)$, $\chi_{440}(133,·)$, $\chi_{440}(263,·)$, $\chi_{440}(397,·)$, $\chi_{440}(21,·)$, $\chi_{440}(23,·)$, $\chi_{440}(153,·)$, $\chi_{440}(287,·)$, $\chi_{440}(417,·)$, $\chi_{440}(419,·)$, $\chi_{440}(43,·)$, $\chi_{440}(177,·)$, $\chi_{440}(307,·)$, $\chi_{440}(309,·)$, $\chi_{440}(439,·)$, $\chi_{440}(67,·)$, $\chi_{440}(197,·)$, $\chi_{440}(199,·)$, $\chi_{440}(329,·)$, $\chi_{440}(331,·)$, $\chi_{440}(87,·)$, $\chi_{440}(89,·)$, $\chi_{440}(219,·)$, $\chi_{440}(221,·)$, $\chi_{440}(351,·)$, $\chi_{440}(353,·)$, $\chi_{440}(109,·)$, $\chi_{440}(111,·)$, $\chi_{440}(241,·)$, $\chi_{440}(243,·)$, $\chi_{440}(373,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{12} + \frac{1}{5} a^{8} - \frac{1}{5} a^{4} + \frac{1}{5}$, $\frac{1}{15} a^{17} - \frac{1}{15} a^{13} + \frac{1}{15} a^{9} - \frac{1}{15} a^{5} + \frac{1}{15} a$, $\frac{1}{45} a^{18} - \frac{16}{45} a^{14} - \frac{14}{45} a^{10} - \frac{1}{45} a^{6} + \frac{16}{45} a^{2}$, $\frac{1}{135} a^{19} - \frac{61}{135} a^{15} - \frac{59}{135} a^{11} - \frac{46}{135} a^{7} - \frac{29}{135} a^{3}$, $\frac{1}{1192725} a^{20} - \frac{17}{405} a^{16} - \frac{43}{405} a^{12} + \frac{58}{405} a^{8} - \frac{163}{405} a^{4} + \frac{6681}{14725}$, $\frac{1}{3578175} a^{21} - \frac{17}{1215} a^{17} - \frac{43}{1215} a^{13} + \frac{463}{1215} a^{9} + \frac{242}{1215} a^{5} - \frac{8044}{44175} a$, $\frac{1}{10734525} a^{22} - \frac{17}{3645} a^{18} + \frac{1172}{3645} a^{14} + \frac{463}{3645} a^{10} + \frac{242}{3645} a^{6} - \frac{8044}{132525} a^{2}$, $\frac{1}{32203575} a^{23} - \frac{17}{10935} a^{19} + \frac{1172}{10935} a^{15} + \frac{4108}{10935} a^{11} - \frac{3403}{10935} a^{7} - \frac{8044}{397575} a^{3}$, $\frac{1}{96610725} a^{24} - \frac{7}{96610725} a^{20} + \frac{929}{32805} a^{16} + \frac{7996}{32805} a^{12} - \frac{1}{32805} a^{8} + \frac{791}{1192725} a^{4} - \frac{32}{14725}$, $\frac{1}{289832175} a^{25} - \frac{7}{289832175} a^{21} + \frac{929}{98415} a^{17} - \frac{24809}{98415} a^{13} - \frac{1}{98415} a^{9} + \frac{791}{3578175} a^{5} - \frac{32}{44175} a$, $\frac{1}{869496525} a^{26} - \frac{7}{869496525} a^{22} + \frac{929}{295245} a^{18} - \frac{24809}{295245} a^{14} + \frac{98414}{295245} a^{10} - \frac{3577384}{10734525} a^{6} + \frac{44143}{132525} a^{2}$, $\frac{1}{2608489575} a^{27} - \frac{7}{2608489575} a^{23} + \frac{929}{885735} a^{19} + \frac{270436}{885735} a^{15} - \frac{196831}{885735} a^{11} - \frac{14311909}{32203575} a^{7} + \frac{44143}{397575} a^{3}$, $\frac{1}{7825468725} a^{28} - \frac{7}{7825468725} a^{24} - \frac{32}{7825468725} a^{20} - \frac{32518}{531441} a^{16} + \frac{106288}{531441} a^{12} - \frac{19321354}{96610725} a^{8} + \frac{238513}{1192725} a^{4} - \frac{2952}{14725}$, $\frac{1}{23476406175} a^{29} - \frac{7}{23476406175} a^{25} - \frac{32}{23476406175} a^{21} - \frac{32518}{1594323} a^{17} - \frac{425153}{1594323} a^{13} + \frac{77289371}{289832175} a^{9} - \frac{954212}{3578175} a^{5} + \frac{11773}{44175} a$, $\frac{1}{70429218525} a^{30} - \frac{7}{70429218525} a^{26} - \frac{32}{70429218525} a^{22} - \frac{32518}{4782969} a^{18} - \frac{2019476}{4782969} a^{14} - \frac{212542804}{869496525} a^{10} - \frac{954212}{10734525} a^{6} + \frac{55948}{132525} a^{2}$, $\frac{1}{211287655575} a^{31} - \frac{7}{211287655575} a^{27} - \frac{32}{211287655575} a^{23} - \frac{32518}{14348907} a^{19} - \frac{2019476}{14348907} a^{15} - \frac{1082039329}{2608489575} a^{11} + \frac{9780313}{32203575} a^{7} + \frac{188473}{397575} a^{3}$

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:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1}{44175} a^{21} + \frac{61126}{44175} a \) (order $40$)
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

$C_2^3\times C_4$ (as 32T34):

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 $C_2^3\times C_4$
Character table for $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{110}) \), \(\Q(\sqrt{-110}) \), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{11})\), \(\Q(i, \sqrt{22})\), \(\Q(\sqrt{2}, \sqrt{11})\), \(\Q(\sqrt{2}, \sqrt{-11})\), \(\Q(\sqrt{-2}, \sqrt{11})\), \(\Q(\sqrt{-2}, \sqrt{-11})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{55})\), \(\Q(i, \sqrt{110})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{55})\), \(\Q(\sqrt{2}, \sqrt{-55})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{55})\), \(\Q(\sqrt{-2}, \sqrt{-55})\), \(\Q(\sqrt{5}, \sqrt{11})\), \(\Q(\sqrt{-5}, \sqrt{11})\), \(\Q(\sqrt{10}, \sqrt{11})\), \(\Q(\sqrt{-10}, \sqrt{11})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{-5}, \sqrt{-11})\), \(\Q(\sqrt{10}, \sqrt{-11})\), \(\Q(\sqrt{-10}, \sqrt{-11})\), \(\Q(\sqrt{5}, \sqrt{22})\), \(\Q(\sqrt{-5}, \sqrt{22})\), \(\Q(\sqrt{10}, \sqrt{22})\), \(\Q(\sqrt{-10}, \sqrt{22})\), \(\Q(\sqrt{5}, \sqrt{-22})\), \(\Q(\sqrt{-5}, \sqrt{-22})\), \(\Q(\sqrt{10}, \sqrt{-22})\), \(\Q(\sqrt{-10}, \sqrt{-22})\), 4.0.968000.5, 4.4.968000.2, 4.0.242000.2, 4.4.15125.1, 4.0.8000.2, 4.4.8000.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 8.0.959512576.1, 8.0.40960000.1, 8.0.599695360000.9, 8.0.2342560000.1, 8.0.599695360000.6, 8.0.599695360000.8, 8.0.599695360000.3, 8.8.599695360000.1, 8.0.599695360000.2, 8.0.37480960000.9, 8.0.599695360000.1, 8.0.599695360000.7, 8.0.599695360000.4, 8.0.37480960000.2, 8.0.599695360000.5, 8.0.14992384000000.7, 8.0.58564000000.2, 8.0.1024000000.2, \(\Q(\zeta_{20})\), 8.0.14992384000000.6, 8.8.937024000000.1, 8.0.64000000.2, \(\Q(\zeta_{40})^+\), 8.0.937024000000.4, 8.0.14992384000000.5, 8.0.1024000000.1, 8.0.64000000.1, 8.0.14992384000000.4, 8.8.14992384000000.2, 8.0.58564000000.3, 8.8.58564000000.1, 8.0.937024000000.3, 8.0.937024000000.2, 8.0.58564000000.1, 8.0.228765625.1, 8.0.937024000000.6, 8.8.14992384000000.1, 8.0.14992384000000.1, 8.8.937024000000.2, 8.0.14992384000000.3, 8.0.937024000000.5, 8.0.14992384000000.2, 8.0.937024000000.1, 16.0.359634524805529600000000.1, 16.0.224771578003456000000000000.3, \(\Q(\zeta_{40})\), 16.0.224771578003456000000000000.8, 16.0.3429742096000000000000.1, 16.0.224771578003456000000000000.6, 16.0.224771578003456000000000000.2, 16.0.224771578003456000000000000.1, 16.16.224771578003456000000000000.1, 16.0.224771578003456000000000000.7, 16.0.878013976576000000000000.1, 16.0.224771578003456000000000000.4, 16.0.224771578003456000000000000.5, 16.0.878013976576000000000000.2, 16.0.224771578003456000000000000.9

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$