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Properties

Label 32.0.15278483419...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{64}\cdot 5^{16}\cdot 13^{24}$
Root discriminant $61.24$
Ramified primes $2, 5, 13$
Class number $1632$ (GRH)
Class group $[4, 408]$ (GRH)
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![6561, 0, 0, 0, -174636, 0, 0, 0, 1488226, 0, 0, 0, -1493310, 0, 0, 0, 703575, 0, 0, 0, -168350, 0, 0, 0, 17931, 0, 0, 0, -189, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 189*x^28 + 17931*x^24 - 168350*x^20 + 703575*x^16 - 1493310*x^12 + 1488226*x^8 - 174636*x^4 + 6561)
 
gp: K = bnfinit(x^32 - 189*x^28 + 17931*x^24 - 168350*x^20 + 703575*x^16 - 1493310*x^12 + 1488226*x^8 - 174636*x^4 + 6561, 1)
 

Normalized defining polynomial

\( x^{32} - 189 x^{28} + 17931 x^{24} - 168350 x^{20} + 703575 x^{16} - 1493310 x^{12} + 1488226 x^{8} - 174636 x^{4} + 6561 \)

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:  \(1527848341996520753686611488433972088668160000000000000000=2^{64}\cdot 5^{16}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.24$
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:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(520=2^{3}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{520}(1,·)$, $\chi_{520}(259,·)$, $\chi_{520}(261,·)$, $\chi_{520}(129,·)$, $\chi_{520}(131,·)$, $\chi_{520}(21,·)$, $\chi_{520}(151,·)$, $\chi_{520}(281,·)$, $\chi_{520}(411,·)$, $\chi_{520}(389,·)$, $\chi_{520}(161,·)$, $\chi_{520}(291,·)$, $\chi_{520}(421,·)$, $\chi_{520}(391,·)$, $\chi_{520}(51,·)$, $\chi_{520}(181,·)$, $\chi_{520}(311,·)$, $\chi_{520}(441,·)$, $\chi_{520}(31,·)$, $\chi_{520}(79,·)$, $\chi_{520}(209,·)$, $\chi_{520}(339,·)$, $\chi_{520}(469,·)$, $\chi_{520}(99,·)$, $\chi_{520}(229,·)$, $\chi_{520}(359,·)$, $\chi_{520}(519,·)$, $\chi_{520}(489,·)$, $\chi_{520}(109,·)$, $\chi_{520}(239,·)$, $\chi_{520}(369,·)$, $\chi_{520}(499,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{6} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{7} - \frac{2}{9} a^{3}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{8} - \frac{2}{9} a^{4}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{9} + \frac{1}{9} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{2}$, $\frac{1}{27} a^{15} - \frac{1}{9} a^{7} + \frac{2}{27} a^{3}$, $\frac{1}{27} a^{16} - \frac{1}{9} a^{8} + \frac{2}{27} a^{4}$, $\frac{1}{27} a^{17} - \frac{1}{9} a^{9} + \frac{2}{27} a^{5}$, $\frac{1}{27} a^{18} - \frac{4}{27} a^{6} + \frac{1}{9} a^{2}$, $\frac{1}{27} a^{19} - \frac{4}{27} a^{7} + \frac{1}{9} a^{3}$, $\frac{1}{243} a^{20} - \frac{4}{243} a^{16} + \frac{2}{81} a^{12} - \frac{4}{243} a^{8} - \frac{80}{243} a^{4} - \frac{1}{3}$, $\frac{1}{243} a^{21} - \frac{4}{243} a^{17} + \frac{2}{81} a^{13} - \frac{4}{243} a^{9} + \frac{1}{243} a^{5} + \frac{1}{3} a$, $\frac{1}{243} a^{22} - \frac{4}{243} a^{18} + \frac{2}{81} a^{14} - \frac{4}{243} a^{10} + \frac{1}{243} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{243} a^{23} - \frac{4}{243} a^{19} - \frac{1}{81} a^{15} - \frac{4}{243} a^{11} + \frac{28}{243} a^{7} + \frac{7}{27} a^{3}$, $\frac{1}{2205711} a^{24} - \frac{94}{2205711} a^{20} - \frac{8674}{735237} a^{16} - \frac{83515}{2205711} a^{12} - \frac{203354}{2205711} a^{8} - \frac{9722}{27231} a^{4} + \frac{113}{9077}$, $\frac{1}{6617133} a^{25} + \frac{8983}{6617133} a^{21} + \frac{101056}{6617133} a^{17} - \frac{274132}{6617133} a^{13} + \frac{495575}{6617133} a^{9} + \frac{38525}{6617133} a^{5} - \frac{8738}{81693} a$, $\frac{1}{6617133} a^{26} + \frac{8983}{6617133} a^{22} + \frac{101056}{6617133} a^{18} - \frac{274132}{6617133} a^{14} - \frac{239662}{6617133} a^{10} - \frac{696712}{6617133} a^{6} + \frac{9416}{81693} a^{2}$, $\frac{1}{6617133} a^{27} + \frac{8983}{6617133} a^{23} + \frac{101056}{6617133} a^{19} - \frac{29053}{6617133} a^{15} - \frac{239662}{6617133} a^{11} + \frac{773762}{6617133} a^{7} - \frac{35291}{245079} a^{3}$, $\frac{1}{8918045113761801} a^{28} + \frac{444268288}{8918045113761801} a^{24} + \frac{252635895178}{146197460881341} a^{20} - \frac{88223751749314}{8918045113761801} a^{16} + \frac{410657025481028}{8918045113761801} a^{12} + \frac{1246647587033}{146197460881341} a^{8} + \frac{338691142811930}{990893901529089} a^{4} - \frac{3257400031968}{12233258043569}$, $\frac{1}{8918045113761801} a^{29} + \frac{444268288}{8918045113761801} a^{25} + \frac{252635895178}{146197460881341} a^{21} - \frac{88223751749314}{8918045113761801} a^{17} + \frac{410657025481028}{8918045113761801} a^{13} + \frac{1246647587033}{146197460881341} a^{9} + \frac{8393175635567}{990893901529089} a^{5} + \frac{2461057947665}{36699774130707} a$, $\frac{1}{26754135341285403} a^{30} - \frac{10384507}{307518797026269} a^{26} + \frac{622614901}{5041291754529} a^{22} + \frac{15040584488420}{922556391078807} a^{18} - \frac{13812384939158}{307518797026269} a^{14} + \frac{261906306788}{5041291754529} a^{10} - \frac{124471034017735}{922556391078807} a^{6} - \frac{54239993589433}{330297967176363} a^{2}$, $\frac{1}{80262406023856209} a^{31} - \frac{61069300}{990893901529089} a^{27} + \frac{421223074825}{438592382644023} a^{23} - \frac{412806036722000}{80262406023856209} a^{19} + \frac{63030923139688}{8918045113761801} a^{15} - \frac{22004545526020}{438592382644023} a^{11} + \frac{2290039960169065}{80262406023856209} a^{7} - \frac{275809270017424}{990893901529089} a^{3}$

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

Class group and class number

$C_{4}\times C_{408}$, which has order $1632$ (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:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{8419316}{235422642321} a^{29} + \frac{42931787272}{6356411342667} a^{25} - \frac{4070042260232}{6356411342667} a^{21} + \frac{37693335580879}{6356411342667} a^{17} - \frac{154304754445609}{6356411342667} a^{13} + \frac{317030159966315}{6356411342667} a^{9} - \frac{295313021679784}{6356411342667} a^{5} + \frac{103132977463}{78474214107} a \) (order $8$)
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:  \( 26450919227224.95 \) (assuming GRH)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{130}) \), \(\Q(\sqrt{-130}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{130})\), \(\Q(i, \sqrt{65})\), \(\Q(\sqrt{2}, \sqrt{65})\), \(\Q(\sqrt{2}, \sqrt{-65})\), \(\Q(\sqrt{-2}, \sqrt{-65})\), \(\Q(\sqrt{-2}, \sqrt{65})\), \(\Q(i, \sqrt{13})\), \(\Q(i, \sqrt{26})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{13})\), \(\Q(\sqrt{2}, \sqrt{-13})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{13})\), \(\Q(\sqrt{-2}, \sqrt{-13})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{10}, \sqrt{13})\), \(\Q(\sqrt{-10}, \sqrt{-13})\), \(\Q(\sqrt{5}, \sqrt{26})\), \(\Q(\sqrt{-5}, \sqrt{-26})\), \(\Q(\sqrt{-10}, \sqrt{13})\), \(\Q(\sqrt{10}, \sqrt{-13})\), \(\Q(\sqrt{-5}, \sqrt{26})\), \(\Q(\sqrt{5}, \sqrt{-26})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-5}, \sqrt{-13})\), \(\Q(\sqrt{10}, \sqrt{26})\), \(\Q(\sqrt{-10}, \sqrt{-26})\), \(\Q(\sqrt{-5}, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{-13})\), \(\Q(\sqrt{-10}, \sqrt{26})\), \(\Q(\sqrt{10}, \sqrt{-26})\), 4.4.3515200.1, 4.0.3515200.1, 4.4.878800.1, 4.0.54925.1, 4.4.35152.1, 4.0.2197.1, 4.4.140608.1, 4.0.140608.2, 8.0.1169858560000.10, 8.0.1871773696.1, 8.0.40960000.1, 8.0.1169858560000.7, 8.0.1169858560000.9, 8.0.4569760000.1, 8.0.1169858560000.1, 8.8.73116160000.2, 8.0.1169858560000.4, 8.0.1169858560000.5, 8.0.1169858560000.8, 8.0.1169858560000.2, 8.0.1169858560000.6, 8.0.73116160000.1, 8.0.1169858560000.3, 8.0.197706096640000.49, 8.0.772289440000.3, 8.0.1235663104.1, 8.0.316329754624.2, 8.8.197706096640000.4, 8.0.12356631040000.2, 8.8.316329754624.1, 8.0.19770609664.2, 8.0.12356631040000.1, 8.0.197706096640000.17, 8.0.316329754624.1, 8.0.19770609664.1, 8.8.197706096640000.10, 8.0.12356631040000.7, 8.8.197706096640000.5, 8.0.12356631040000.4, 8.0.12356631040000.6, 8.0.197706096640000.43, 8.0.197706096640000.47, 8.0.12356631040000.3, 8.8.12356631040000.1, 8.0.12356631040000.5, 8.8.772289440000.1, 8.0.3016755625.1, 8.0.197706096640000.20, 8.0.197706096640000.72, 8.0.772289440000.2, 8.0.772289440000.1, 16.0.1368569050405273600000000.1, 16.0.39087700648625019289600000000.4, 16.0.100064513660480049381376.1, 16.0.39087700648625019289600000000.1, 16.0.39087700648625019289600000000.8, 16.0.39087700648625019289600000000.2, 16.0.596430979135513600000000.1, 16.16.39087700648625019289600000000.1, 16.0.152686330658691481600000000.2, 16.0.39087700648625019289600000000.5, 16.0.39087700648625019289600000000.9, 16.0.39087700648625019289600000000.3, 16.0.39087700648625019289600000000.7, 16.0.152686330658691481600000000.1, 16.0.39087700648625019289600000000.6

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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{16}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$