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Properties

Label 32.0.54568201713...0000.5
Degree $32$
Signature $[0, 16]$
Discriminant $2^{124}\cdot 3^{16}\cdot 5^{24}$
Root discriminant $84.97$
Ramified primes $2, 3, 5$
Class number Not computed
Class group Not computed
Galois group $C_4\times C_8$ (as 32T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![688747536, 0, -1836660096, 0, 4132485216, 0, -9081263808, 0, 19892899512, 0, -8899865280, 0, 2949615648, 0, -867941568, 0, 232863012, 0, -38631168, 0, 5423760, 0, -680400, 0, 73710, 0, -5616, 0, 396, 0, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 24*x^30 + 396*x^28 - 5616*x^26 + 73710*x^24 - 680400*x^22 + 5423760*x^20 - 38631168*x^18 + 232863012*x^16 - 867941568*x^14 + 2949615648*x^12 - 8899865280*x^10 + 19892899512*x^8 - 9081263808*x^6 + 4132485216*x^4 - 1836660096*x^2 + 688747536)
 
gp: K = bnfinit(x^32 - 24*x^30 + 396*x^28 - 5616*x^26 + 73710*x^24 - 680400*x^22 + 5423760*x^20 - 38631168*x^18 + 232863012*x^16 - 867941568*x^14 + 2949615648*x^12 - 8899865280*x^10 + 19892899512*x^8 - 9081263808*x^6 + 4132485216*x^4 - 1836660096*x^2 + 688747536, 1)
 

Normalized defining polynomial

\( x^{32} - 24 x^{30} + 396 x^{28} - 5616 x^{26} + 73710 x^{24} - 680400 x^{22} + 5423760 x^{20} - 38631168 x^{18} + 232863012 x^{16} - 867941568 x^{14} + 2949615648 x^{12} - 8899865280 x^{10} + 19892899512 x^{8} - 9081263808 x^{6} + 4132485216 x^{4} - 1836660096 x^{2} + 688747536 \)

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:  \(54568201713507127370225565301626372096000000000000000000000000=2^{124}\cdot 3^{16}\cdot 5^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.97$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(480=2^{5}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{480}(1,·)$, $\chi_{480}(389,·)$, $\chi_{480}(269,·)$, $\chi_{480}(149,·)$, $\chi_{480}(409,·)$, $\chi_{480}(413,·)$, $\chi_{480}(289,·)$, $\chi_{480}(293,·)$, $\chi_{480}(49,·)$, $\chi_{480}(169,·)$, $\chi_{480}(173,·)$, $\chi_{480}(29,·)$, $\chi_{480}(433,·)$, $\chi_{480}(53,·)$, $\chi_{480}(73,·)$, $\chi_{480}(313,·)$, $\chi_{480}(317,·)$, $\chi_{480}(437,·)$, $\chi_{480}(193,·)$, $\chi_{480}(197,·)$, $\chi_{480}(457,·)$, $\chi_{480}(77,·)$, $\chi_{480}(461,·)$, $\chi_{480}(337,·)$, $\chi_{480}(341,·)$, $\chi_{480}(217,·)$, $\chi_{480}(221,·)$, $\chi_{480}(97,·)$, $\chi_{480}(101,·)$, $\chi_{480}(361,·)$, $\chi_{480}(241,·)$, $\chi_{480}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{162} a^{8}$, $\frac{1}{162} a^{9}$, $\frac{1}{486} a^{10}$, $\frac{1}{486} a^{11}$, $\frac{1}{1458} a^{12}$, $\frac{1}{1458} a^{13}$, $\frac{1}{4374} a^{14}$, $\frac{1}{4374} a^{15}$, $\frac{1}{26244} a^{16}$, $\frac{1}{26244} a^{17}$, $\frac{1}{2440692} a^{18} + \frac{5}{271188} a^{16} + \frac{2}{67797} a^{14} - \frac{1}{22599} a^{12} + \frac{1}{15066} a^{10} - \frac{13}{5022} a^{8} + \frac{11}{837} a^{6} + \frac{10}{279} a^{4} - \frac{5}{93} a^{2} - \frac{13}{31}$, $\frac{1}{2440692} a^{19} + \frac{5}{271188} a^{17} + \frac{2}{67797} a^{15} - \frac{1}{22599} a^{13} + \frac{1}{15066} a^{11} - \frac{13}{5022} a^{9} + \frac{11}{837} a^{7} + \frac{10}{279} a^{5} - \frac{5}{93} a^{3} - \frac{13}{31} a$, $\frac{1}{7322076} a^{20} + \frac{1}{5022} a^{10} + \frac{9}{31}$, $\frac{1}{7322076} a^{21} + \frac{1}{5022} a^{11} + \frac{9}{31} a$, $\frac{1}{21966228} a^{22} + \frac{1}{15066} a^{12} + \frac{3}{31} a^{2}$, $\frac{1}{21966228} a^{23} + \frac{1}{15066} a^{13} + \frac{3}{31} a^{3}$, $\frac{1}{131797368} a^{24} - \frac{7}{67797} a^{14} - \frac{11}{279} a^{4}$, $\frac{1}{131797368} a^{25} - \frac{7}{67797} a^{15} - \frac{11}{279} a^{5}$, $\frac{1}{163198486318104} a^{26} + \frac{3173}{877411216764} a^{24} + \frac{85069}{9066582573228} a^{22} - \frac{85657}{1511097095538} a^{20} + \frac{26513}{167899677282} a^{18} + \frac{411907}{335799354564} a^{16} - \frac{191051}{18655519698} a^{14} - \frac{31366}{9327759849} a^{12} + \frac{1048273}{2072835522} a^{10} + \frac{3841175}{2072835522} a^{8} - \frac{5143468}{345472587} a^{6} + \frac{3127171}{115157529} a^{4} - \frac{1006672}{38385843} a^{2} - \frac{4928301}{12795281}$, $\frac{1}{163198486318104} a^{27} + \frac{3173}{877411216764} a^{25} + \frac{85069}{9066582573228} a^{23} - \frac{85657}{1511097095538} a^{21} + \frac{26513}{167899677282} a^{19} + \frac{411907}{335799354564} a^{17} - \frac{191051}{18655519698} a^{15} - \frac{31366}{9327759849} a^{13} + \frac{1048273}{2072835522} a^{11} + \frac{3841175}{2072835522} a^{9} - \frac{5143468}{345472587} a^{7} + \frac{3127171}{115157529} a^{5} - \frac{1006672}{38385843} a^{3} - \frac{4928301}{12795281} a$, $\frac{1}{15177459227583672} a^{28} + \frac{5}{1686384358620408} a^{26} - \frac{1353545}{1686384358620408} a^{24} - \frac{690515}{46844009961678} a^{22} - \frac{4501075}{93688019923356} a^{20} + \frac{1974767}{31229339974452} a^{18} + \frac{44767237}{10409779991484} a^{16} - \frac{122974903}{1734963331914} a^{14} + \frac{3698195}{96386851773} a^{12} + \frac{181713821}{192773703546} a^{10} - \frac{35684806}{32128950591} a^{8} + \frac{174213533}{10709650197} a^{6} - \frac{10076623}{396653711} a^{4} + \frac{125886043}{1189961133} a^{2} + \frac{9949030}{396653711}$, $\frac{1}{15177459227583672} a^{29} + \frac{5}{1686384358620408} a^{27} - \frac{1353545}{1686384358620408} a^{25} - \frac{690515}{46844009961678} a^{23} - \frac{4501075}{93688019923356} a^{21} + \frac{1974767}{31229339974452} a^{19} + \frac{44767237}{10409779991484} a^{17} - \frac{122974903}{1734963331914} a^{15} + \frac{3698195}{96386851773} a^{13} + \frac{181713821}{192773703546} a^{11} - \frac{35684806}{32128950591} a^{9} + \frac{174213533}{10709650197} a^{7} - \frac{10076623}{396653711} a^{5} + \frac{125886043}{1189961133} a^{3} + \frac{9949030}{396653711} a$, $\frac{1}{45532377682751016} a^{30} + \frac{57208}{23422004980839} a^{20} - \frac{6083452}{10709650197} a^{10} - \frac{130294233}{396653711}$, $\frac{1}{45532377682751016} a^{31} + \frac{57208}{23422004980839} a^{21} - \frac{6083452}{10709650197} a^{11} - \frac{130294233}{396653711} a$

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:  \( -\frac{861250}{5691547210343877} a^{30} + \frac{689000}{210798044827551} a^{28} - \frac{32469125}{632394134482653} a^{26} + \frac{1196091895}{1686384358620408} a^{24} - \frac{644215000}{70266014942517} a^{22} + \frac{600808000}{7807334993613} a^{20} - \frac{4556873750}{7807334993613} a^{18} + \frac{10328110000}{2602444997871} a^{16} - \frac{19039316666}{867481665957} a^{14} + \frac{15045004000}{289160555319} a^{12} - \frac{5231921500}{32128950591} a^{10} + \frac{468520000}{1189961133} a^{8} - \frac{1925066000}{10709650197} a^{6} - \frac{16822240718}{3569883399} a^{4} - \frac{14469000}{396653711} a^{2} + \frac{5512000}{396653711} \) (order $10$)
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

$C_4\times C_8$ (as 32T43):

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 $C_4\times C_8$
Character table for $C_4\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{5})\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.8000.2, 4.4.51200.1, \(\Q(\zeta_{16})^+\), 4.0.256000.2, 4.0.256000.4, 8.0.64000000.2, 8.8.2621440000.1, 8.0.65536000000.1, 8.8.2717908992000000.1, 8.8.2717908992000000.2, 8.0.108716359680000.13, 8.0.173946175488.1, 16.0.4294967296000000000000.1, 16.16.7387029288794456064000000000000.1, 16.0.11819246862071129702400000000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{32}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
2Data not computed
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
5Data not computed