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Properties

Label 32.0.59816430901...0000.4
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{24}$
Root discriminant $79.30$
Ramified primes $2, 3, 5, 13$
Class number $327680$ (GRH)
Class group $[8, 8, 8, 8, 80]$ (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![32041, 0, 3838047, 0, 35183233, 0, 145473897, 0, 350875173, 0, 547864098, 0, 583652697, 0, 436175237, 0, 231662344, 0, 87672250, 0, 23500815, 0, 4400894, 0, 563517, 0, 47810, 0, 2550, 0, 77, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 77*x^30 + 2550*x^28 + 47810*x^26 + 563517*x^24 + 4400894*x^22 + 23500815*x^20 + 87672250*x^18 + 231662344*x^16 + 436175237*x^14 + 583652697*x^12 + 547864098*x^10 + 350875173*x^8 + 145473897*x^6 + 35183233*x^4 + 3838047*x^2 + 32041)
 
gp: K = bnfinit(x^32 + 77*x^30 + 2550*x^28 + 47810*x^26 + 563517*x^24 + 4400894*x^22 + 23500815*x^20 + 87672250*x^18 + 231662344*x^16 + 436175237*x^14 + 583652697*x^12 + 547864098*x^10 + 350875173*x^8 + 145473897*x^6 + 35183233*x^4 + 3838047*x^2 + 32041, 1)
 

Normalized defining polynomial

\( x^{32} + 77 x^{30} + 2550 x^{28} + 47810 x^{26} + 563517 x^{24} + 4400894 x^{22} + 23500815 x^{20} + 87672250 x^{18} + 231662344 x^{16} + 436175237 x^{14} + 583652697 x^{12} + 547864098 x^{10} + 350875173 x^{8} + 145473897 x^{6} + 35183233 x^{4} + 3838047 x^{2} + 32041 \)

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:  \(5981643090147991811559885370844487936000000000000000000000000=2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.30$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(780=2^{2}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{780}(1,·)$, $\chi_{780}(259,·)$, $\chi_{780}(391,·)$, $\chi_{780}(649,·)$, $\chi_{780}(281,·)$, $\chi_{780}(623,·)$, $\chi_{780}(287,·)$, $\chi_{780}(161,·)$, $\chi_{780}(677,·)$, $\chi_{780}(551,·)$, $\chi_{780}(307,·)$, $\chi_{780}(53,·)$, $\chi_{780}(697,·)$, $\chi_{780}(571,·)$, $\chi_{780}(671,·)$, $\chi_{780}(181,·)$, $\chi_{780}(577,·)$, $\chi_{780}(73,·)$, $\chi_{780}(77,·)$, $\chi_{780}(79,·)$, $\chi_{780}(467,·)$, $\chi_{780}(469,·)$, $\chi_{780}(343,·)$, $\chi_{780}(463,·)$, $\chi_{780}(733,·)$, $\chi_{780}(187,·)$, $\chi_{780}(359,·)$, $\chi_{780}(233,·)$, $\chi_{780}(749,·)$, $\chi_{780}(239,·)$, $\chi_{780}(629,·)$, $\chi_{780}(443,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{4} a^{20} - \frac{1}{2} a^{18} + \frac{1}{4} a^{16} - \frac{1}{2} a^{14} + \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{21} - \frac{1}{2} a^{19} + \frac{1}{4} a^{17} - \frac{1}{2} a^{15} + \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{22} + \frac{1}{4} a^{18} + \frac{1}{4} a^{12} + \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{23} + \frac{1}{4} a^{19} + \frac{1}{4} a^{13} + \frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{12} a^{24} + \frac{1}{12} a^{20} - \frac{1}{3} a^{18} - \frac{1}{3} a^{16} + \frac{5}{12} a^{14} + \frac{1}{12} a^{12} - \frac{1}{2} a^{10} - \frac{5}{12} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} + \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{12} a^{25} + \frac{1}{12} a^{21} - \frac{1}{3} a^{19} - \frac{1}{3} a^{17} + \frac{5}{12} a^{15} + \frac{1}{12} a^{13} - \frac{1}{2} a^{11} - \frac{5}{12} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} + \frac{1}{6} a^{3} + \frac{1}{3} a$, $\frac{1}{12} a^{26} + \frac{1}{12} a^{22} - \frac{1}{12} a^{20} + \frac{1}{6} a^{18} - \frac{1}{3} a^{16} - \frac{5}{12} a^{14} - \frac{1}{2} a^{12} - \frac{1}{6} a^{10} - \frac{1}{2} a^{8} + \frac{1}{4} a^{6} - \frac{1}{12} a^{4} + \frac{1}{12} a^{2} + \frac{1}{4}$, $\frac{1}{12} a^{27} + \frac{1}{12} a^{23} - \frac{1}{12} a^{21} + \frac{1}{6} a^{19} - \frac{1}{3} a^{17} - \frac{5}{12} a^{15} - \frac{1}{2} a^{13} - \frac{1}{6} a^{11} - \frac{1}{2} a^{9} + \frac{1}{4} a^{7} - \frac{1}{12} a^{5} + \frac{1}{12} a^{3} + \frac{1}{4} a$, $\frac{1}{1249524} a^{28} - \frac{10427}{1249524} a^{26} + \frac{1005}{138836} a^{24} - \frac{2491}{138836} a^{22} + \frac{13801}{416508} a^{20} - \frac{90673}{1249524} a^{18} - \frac{268499}{1249524} a^{16} - \frac{1135}{1249524} a^{14} - \frac{166537}{416508} a^{12} - \frac{465665}{1249524} a^{10} - \frac{222667}{1249524} a^{8} - \frac{115207}{1249524} a^{6} + \frac{65417}{416508} a^{4} + \frac{185}{20484} a^{2} + \frac{185927}{1249524}$, $\frac{1}{223664796} a^{29} - \frac{497210}{55916199} a^{27} - \frac{1334789}{37277466} a^{25} - \frac{1812341}{74554932} a^{23} + \frac{2181440}{18638733} a^{21} - \frac{13064575}{55916199} a^{19} - \frac{21387128}{55916199} a^{17} - \frac{58312255}{223664796} a^{15} - \frac{20020085}{74554932} a^{13} - \frac{89598377}{223664796} a^{11} + \frac{32681465}{223664796} a^{9} - \frac{31139513}{111832398} a^{7} - \frac{28326545}{74554932} a^{5} - \frac{516275}{1833318} a^{3} + \frac{92442449}{223664796} a$, $\frac{1}{13179018864655803434977632782495484} a^{30} - \frac{734068686988528808461912085}{4393006288218601144992544260831828} a^{28} - \frac{494501597501919978936780730436833}{13179018864655803434977632782495484} a^{26} - \frac{45492882883737085313813569372799}{1464335429406200381664181420277276} a^{24} - \frac{105272212107121148577212771564173}{4393006288218601144992544260831828} a^{22} - \frac{25465064036498768436548395413037}{13179018864655803434977632782495484} a^{20} + \frac{1204734676212073857255877332466553}{13179018864655803434977632782495484} a^{18} - \frac{6124571936170867908366656046542129}{13179018864655803434977632782495484} a^{16} - \frac{1755014279729730323062842040882013}{13179018864655803434977632782495484} a^{14} + \frac{4406479423704200382252348768093547}{13179018864655803434977632782495484} a^{12} - \frac{3496456389611442882873877895417}{21928483967813316863523515445084} a^{10} + \frac{159286756166919150198880563299295}{1464335429406200381664181420277276} a^{8} + \frac{6123808992447710384745170251830325}{13179018864655803434977632782495484} a^{6} + \frac{3940482102857693453072943273794237}{13179018864655803434977632782495484} a^{4} - \frac{19938402811371735564618062669673}{1464335429406200381664181420277276} a^{2} - \frac{651648152977575530783415882715}{18406450928290228261141945226949}$, $\frac{1}{13179018864655803434977632782495484} a^{31} - \frac{11026034207830193715638441}{6589509432327901717488816391247742} a^{29} - \frac{437472775826313735269583449002325}{13179018864655803434977632782495484} a^{27} - \frac{33076022049952060611701590354629}{1464335429406200381664181420277276} a^{25} - \frac{32968008800295474997431839616098}{366083857351550095416045355069319} a^{23} + \frac{258551566818416126933255633656649}{3294754716163950858744408195623871} a^{21} - \frac{967083169064899129055346799594549}{4393006288218601144992544260831828} a^{19} + \frac{512502717684631303790537385957926}{1098251572054650286248136065207957} a^{17} - \frac{1842015309169492523557721976345515}{6589509432327901717488816391247742} a^{15} + \frac{1529780097168138044423948066556179}{6589509432327901717488816391247742} a^{13} - \frac{969791925149961462367045468107}{2436498218645924095947057271676} a^{11} + \frac{1297855024080672307658416423797895}{13179018864655803434977632782495484} a^{9} - \frac{4453378979342806998810080421473275}{13179018864655803434977632782495484} a^{7} - \frac{7925668413056317123626106094189}{23161720324526895316305154275036} a^{5} + \frac{2088404432147273673022745643369623}{6589509432327901717488816391247742} a^{3} + \frac{2242581396564286199459946733438117}{6589509432327901717488816391247742} a$

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

Class group and class number

$C_{8}\times C_{8}\times C_{8}\times C_{8}\times C_{80}$, which has order $327680$ (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:  \( \frac{43682034905041405}{22711506177545765478} a^{31} + \frac{3317158735559980025}{22711506177545765478} a^{29} + \frac{107871984959297161915}{22711506177545765478} a^{27} + \frac{658064464017861030793}{7570502059181921826} a^{25} + \frac{7509301460755185712015}{7570502059181921826} a^{23} + \frac{168464247274183885950875}{22711506177545765478} a^{21} + \frac{283074301643995409285335}{7570502059181921826} a^{19} + \frac{979548617429338639682375}{7570502059181921826} a^{17} + \frac{7048471641891929638503323}{22711506177545765478} a^{15} + \frac{11722851654904466996253865}{22711506177545765478} a^{13} + \frac{1486089326830235390300405}{2523500686393973942} a^{11} + \frac{10200541994189800948268935}{22711506177545765478} a^{9} + \frac{4944073271473062848767115}{22711506177545765478} a^{7} + \frac{1375506047999436399034841}{22711506177545765478} a^{5} + \frac{170411637446790306201145}{22711506177545765478} a^{3} + \frac{837659191729676649625}{11355753088772882739} a \) (order $4$)
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:  \( 1189280875603.7385 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(i, \sqrt{65})\), 4.4.274625.1, 4.0.4394000.2, \(\Q(i, \sqrt{13})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-5}, \sqrt{-13})\), 4.4.274625.2, 4.0.4394000.1, \(\Q(\sqrt{-5}, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{-13})\), 4.0.316368.2, 4.4.19773.1, 4.0.7909200.1, 4.4.494325.1, 4.0.18000.1, \(\Q(\zeta_{15})^+\), 4.0.3042000.1, 4.4.190125.1, 8.0.19307236000000.1, 8.0.4569760000.1, 8.0.19307236000000.4, 8.8.75418890625.1, 8.0.19307236000000.2, 8.0.19307236000000.5, 8.0.19307236000000.3, 8.0.100088711424.1, 8.0.62555444640000.62, 8.0.324000000.1, 8.0.9253764000000.8, 8.0.62555444640000.63, 8.8.244357205625.1, 8.0.9253764000000.3, 8.8.36147515625.1, 8.0.62555444640000.68, 8.0.62555444640000.43, 8.0.9253764000000.1, 8.0.9253764000000.7, 16.0.372769361959696000000000000.1, 16.0.3913183654108104729600000000.8, 16.0.85632148167696000000000000.5, 16.0.2445739783817565456000000000000.13, 16.16.37319027463036582275390625.1, 16.0.2445739783817565456000000000000.15, 16.0.2445739783817565456000000000000.16

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
13Data not computed