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Properties

Label 32.0.34351813240...6016.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{48}\cdot 3^{16}\cdot 17^{28}$
Root discriminant $58.45$
Ramified primes $2, 3, 17$
Class number $360$ (GRH)
Class group $[3, 120]$ (GRH)
Galois group $C_2^2\times C_8$ (as 32T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, 0, -1179648, 0, 17793024, 0, -54362112, 0, 110481408, 0, -126291968, 0, 101765120, 0, -57272832, 0, 24284672, 0, -7766016, 0, 1935424, 0, -372416, 0, 55936, 0, -6344, 0, 536, 0, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 30*x^30 + 536*x^28 - 6344*x^26 + 55936*x^24 - 372416*x^22 + 1935424*x^20 - 7766016*x^18 + 24284672*x^16 - 57272832*x^14 + 101765120*x^12 - 126291968*x^10 + 110481408*x^8 - 54362112*x^6 + 17793024*x^4 - 1179648*x^2 + 65536)
 
gp: K = bnfinit(x^32 - 30*x^30 + 536*x^28 - 6344*x^26 + 55936*x^24 - 372416*x^22 + 1935424*x^20 - 7766016*x^18 + 24284672*x^16 - 57272832*x^14 + 101765120*x^12 - 126291968*x^10 + 110481408*x^8 - 54362112*x^6 + 17793024*x^4 - 1179648*x^2 + 65536, 1)
 

Normalized defining polynomial

\( x^{32} - 30 x^{30} + 536 x^{28} - 6344 x^{26} + 55936 x^{24} - 372416 x^{22} + 1935424 x^{20} - 7766016 x^{18} + 24284672 x^{16} - 57272832 x^{14} + 101765120 x^{12} - 126291968 x^{10} + 110481408 x^{8} - 54362112 x^{6} + 17793024 x^{4} - 1179648 x^{2} + 65536 \)

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:  \(343518132400968646195554032900431766403471261479800406016=2^{48}\cdot 3^{16}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.45$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(408=2^{3}\cdot 3\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{408}(1,·)$, $\chi_{408}(259,·)$, $\chi_{408}(257,·)$, $\chi_{408}(137,·)$, $\chi_{408}(395,·)$, $\chi_{408}(145,·)$, $\chi_{408}(19,·)$, $\chi_{408}(25,·)$, $\chi_{408}(281,·)$, $\chi_{408}(155,·)$, $\chi_{408}(161,·)$, $\chi_{408}(35,·)$, $\chi_{408}(49,·)$, $\chi_{408}(169,·)$, $\chi_{408}(43,·)$, $\chi_{408}(307,·)$, $\chi_{408}(305,·)$, $\chi_{408}(179,·)$, $\chi_{408}(185,·)$, $\chi_{408}(59,·)$, $\chi_{408}(67,·)$, $\chi_{408}(331,·)$, $\chi_{408}(83,·)$, $\chi_{408}(203,·)$, $\chi_{408}(89,·)$, $\chi_{408}(353,·)$, $\chi_{408}(355,·)$, $\chi_{408}(361,·)$, $\chi_{408}(115,·)$, $\chi_{408}(121,·)$, $\chi_{408}(217,·)$, $\chi_{408}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{16384} a^{24} - \frac{1}{2048} a^{18} + \frac{1}{256} a^{12} - \frac{1}{32} a^{6} + \frac{1}{4}$, $\frac{1}{16384} a^{25} - \frac{1}{2048} a^{19} + \frac{1}{256} a^{13} - \frac{1}{32} a^{7} + \frac{1}{4} a$, $\frac{1}{32768} a^{26} - \frac{1}{4096} a^{20} + \frac{1}{512} a^{14} - \frac{1}{64} a^{8} + \frac{1}{8} a^{2}$, $\frac{1}{32768} a^{27} - \frac{1}{4096} a^{21} + \frac{1}{512} a^{15} - \frac{1}{64} a^{9} + \frac{1}{8} a^{3}$, $\frac{1}{5879431168} a^{28} + \frac{3023}{226131968} a^{26} + \frac{7889}{367464448} a^{24} + \frac{3975}{734928896} a^{22} + \frac{84017}{367464448} a^{20} + \frac{4629}{11483264} a^{18} - \frac{132151}{91866112} a^{16} + \frac{62195}{45933056} a^{14} + \frac{33355}{5741632} a^{12} - \frac{114209}{11483264} a^{10} - \frac{47527}{5741632} a^{8} + \frac{30317}{717704} a^{6} - \frac{138651}{1435408} a^{4} - \frac{130001}{717704} a^{2} - \frac{34733}{89713}$, $\frac{1}{5879431168} a^{29} + \frac{3023}{226131968} a^{27} + \frac{7889}{367464448} a^{25} + \frac{3975}{734928896} a^{23} + \frac{84017}{367464448} a^{21} + \frac{4629}{11483264} a^{19} - \frac{132151}{91866112} a^{17} + \frac{62195}{45933056} a^{15} + \frac{33355}{5741632} a^{13} - \frac{114209}{11483264} a^{11} - \frac{47527}{5741632} a^{9} + \frac{30317}{717704} a^{7} - \frac{138651}{1435408} a^{5} - \frac{130001}{717704} a^{3} - \frac{34733}{89713} a$, $\frac{1}{575751097677376707493888} a^{30} + \frac{7900196601225}{143937774419344176873472} a^{28} - \frac{999041555612202613}{71968887209672088436736} a^{26} - \frac{1338024522544717873}{71968887209672088436736} a^{24} + \frac{3835392211117073523}{17992221802418022109184} a^{22} + \frac{3694594152601151133}{8996110901209011054592} a^{20} + \frac{4996032877804513497}{8996110901209011054592} a^{18} - \frac{147626888757663837}{2249027725302252763648} a^{16} - \frac{3820195192929595923}{1124513862651126381824} a^{14} + \frac{4830698173074491843}{1124513862651126381824} a^{12} + \frac{1838877410860332189}{281128465662781595456} a^{10} + \frac{1778620435172716663}{140564232831390797728} a^{8} + \frac{3463142646295211545}{140564232831390797728} a^{6} - \frac{3220148525250131219}{35141058207847699432} a^{4} + \frac{178917159753353623}{1351579161840296132} a^{2} + \frac{889741426692540796}{4392632275980962429}$, $\frac{1}{575751097677376707493888} a^{31} + \frac{7900196601225}{143937774419344176873472} a^{29} - \frac{999041555612202613}{71968887209672088436736} a^{27} - \frac{1338024522544717873}{71968887209672088436736} a^{25} + \frac{3835392211117073523}{17992221802418022109184} a^{23} + \frac{3694594152601151133}{8996110901209011054592} a^{21} + \frac{4996032877804513497}{8996110901209011054592} a^{19} - \frac{147626888757663837}{2249027725302252763648} a^{17} - \frac{3820195192929595923}{1124513862651126381824} a^{15} + \frac{4830698173074491843}{1124513862651126381824} a^{13} + \frac{1838877410860332189}{281128465662781595456} a^{11} + \frac{1778620435172716663}{140564232831390797728} a^{9} + \frac{3463142646295211545}{140564232831390797728} a^{7} - \frac{3220148525250131219}{35141058207847699432} a^{5} + \frac{178917159753353623}{1351579161840296132} a^{3} + \frac{889741426692540796}{4392632275980962429} a$

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

Class group and class number

$C_{3}\times C_{120}$, which has order $360$ (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:  \( -\frac{666255728058537507}{575751097677376707493888} a^{30} + \frac{9958026200414712059}{287875548838688353746944} a^{28} - \frac{44375142701574773463}{71968887209672088436736} a^{26} + \frac{523676182392255393901}{71968887209672088436736} a^{24} - \frac{2302009833061492251667}{35984443604836044218368} a^{22} + \frac{3817758492490219804039}{8996110901209011054592} a^{20} - \frac{19760992010027759365005}{8996110901209011054592} a^{18} + \frac{3033538926927053056667}{346004265431115809792} a^{16} - \frac{457256879154747983161}{16783788994792931072} a^{14} + \frac{71601600844042490490521}{1124513862651126381824} a^{12} - \frac{62914914062269543384987}{562256931325563190912} a^{10} + \frac{19158178071443160895945}{140564232831390797728} a^{8} - \frac{16424218762316464539945}{140564232831390797728} a^{6} + \frac{3823365820404516748143}{70282116415695398864} a^{4} - \frac{322367663821497420007}{17570529103923849716} a^{2} + \frac{10679250748987617087}{8785264551961924858} \) (order $6$)
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:  \( 12886153151831.549 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_8$ (as 32T37):

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 $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{102}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\sqrt{-3}, \sqrt{-34})\), \(\Q(\sqrt{-2}, \sqrt{17})\), \(\Q(\sqrt{-2}, \sqrt{-51})\), \(\Q(\sqrt{6}, \sqrt{17})\), \(\Q(\sqrt{6}, \sqrt{-34})\), 4.4.4913.1, 4.0.44217.1, 4.0.314432.2, 4.4.2829888.2, 8.0.27710263296.5, 8.0.1955143089.1, 8.0.8008266092544.6, 8.0.98867482624.1, 8.0.8008266092544.3, 8.8.8008266092544.1, 8.0.8008266092544.1, \(\Q(\zeta_{17})^+\), 8.0.33237432513.1, 8.0.1680747204608.1, 8.8.136140523573248.1, 16.0.64132325808989945972391936.2, 16.0.1104726920056229495169.1, 16.0.18534242158798094386021269504.6, 16.0.2824911165797606216433664.1, 16.0.18534242158798094386021269504.4, 16.16.18534242158798094386021269504.1, 16.0.18534242158798094386021269504.2

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
3Data not computed
17Data not computed