Properties

Label 32.32.1094872516...7168.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{48}\cdot 97^{31}$
Root discriminant $237.81$
Ramified primes $2, 97$
Class number Not computed
Class group Not computed
Galois group $C_{32}$ (as 32T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6356992, 0, -820051968, 0, 17972805632, 0, -153945686016, 0, 609862000640, 0, -1149333823488, 0, 1064307168256, 0, -547571276288, 0, 170321943040, 0, -33633640320, 0, 4325759232, 0, -365958496, 0, 20303264, 0, -724784, 0, 15908, 0, -194, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 194*x^30 + 15908*x^28 - 724784*x^26 + 20303264*x^24 - 365958496*x^22 + 4325759232*x^20 - 33633640320*x^18 + 170321943040*x^16 - 547571276288*x^14 + 1064307168256*x^12 - 1149333823488*x^10 + 609862000640*x^8 - 153945686016*x^6 + 17972805632*x^4 - 820051968*x^2 + 6356992)
 
gp: K = bnfinit(x^32 - 194*x^30 + 15908*x^28 - 724784*x^26 + 20303264*x^24 - 365958496*x^22 + 4325759232*x^20 - 33633640320*x^18 + 170321943040*x^16 - 547571276288*x^14 + 1064307168256*x^12 - 1149333823488*x^10 + 609862000640*x^8 - 153945686016*x^6 + 17972805632*x^4 - 820051968*x^2 + 6356992, 1)
 

Normalized defining polynomial

\( x^{32} - 194 x^{30} + 15908 x^{28} - 724784 x^{26} + 20303264 x^{24} - 365958496 x^{22} + 4325759232 x^{20} - 33633640320 x^{18} + 170321943040 x^{16} - 547571276288 x^{14} + 1064307168256 x^{12} - 1149333823488 x^{10} + 609862000640 x^{8} - 153945686016 x^{6} + 17972805632 x^{4} - 820051968 x^{2} + 6356992 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[32, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10948725162062396375893585077651059129669167570225987349215648531823007367168=2^{48}\cdot 97^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $237.81$
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:  $2, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(776=2^{3}\cdot 97\)
Dirichlet character group:    $\lbrace$$\chi_{776}(1,·)$, $\chi_{776}(515,·)$, $\chi_{776}(139,·)$, $\chi_{776}(273,·)$, $\chi_{776}(131,·)$, $\chi_{776}(729,·)$, $\chi_{776}(33,·)$, $\chi_{776}(241,·)$, $\chi_{776}(51,·)$, $\chi_{776}(313,·)$, $\chi_{776}(699,·)$, $\chi_{776}(193,·)$, $\chi_{776}(67,·)$, $\chi_{776}(161,·)$, $\chi_{776}(707,·)$, $\chi_{776}(531,·)$, $\chi_{776}(659,·)$, $\chi_{776}(563,·)$, $\chi_{776}(697,·)$, $\chi_{776}(473,·)$, $\chi_{776}(731,·)$, $\chi_{776}(627,·)$, $\chi_{776}(609,·)$, $\chi_{776}(443,·)$, $\chi_{776}(651,·)$, $\chi_{776}(657,·)$, $\chi_{776}(105,·)$, $\chi_{776}(451,·)$, $\chi_{776}(497,·)$, $\chi_{776}(19,·)$, $\chi_{776}(89,·)$, $\chi_{776}(361,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{249856} a^{24} + \frac{15}{124928} a^{22} - \frac{3}{15616} a^{20} + \frac{9}{15616} a^{18} + \frac{7}{7808} a^{16} - \frac{19}{7808} a^{14} - \frac{5}{976} a^{12} + \frac{11}{1952} a^{10} + \frac{5}{488} a^{8} - \frac{7}{488} a^{6} + \frac{9}{122} a^{4} + \frac{9}{61} a^{2} + \frac{5}{61}$, $\frac{1}{249856} a^{25} + \frac{15}{124928} a^{23} - \frac{3}{15616} a^{21} + \frac{9}{15616} a^{19} + \frac{7}{7808} a^{17} - \frac{19}{7808} a^{15} - \frac{5}{976} a^{13} + \frac{11}{1952} a^{11} + \frac{5}{488} a^{9} - \frac{7}{488} a^{7} + \frac{9}{122} a^{5} + \frac{9}{61} a^{3} + \frac{5}{61} a$, $\frac{1}{499712} a^{26} + \frac{7}{124928} a^{22} + \frac{15}{62464} a^{20} - \frac{3}{7808} a^{18} + \frac{15}{15616} a^{16} + \frac{21}{7808} a^{14} + \frac{3}{1952} a^{12} + \frac{7}{488} a^{10} + \frac{13}{488} a^{8} + \frac{1}{488} a^{6} - \frac{2}{61} a^{4} - \frac{21}{122} a^{2} - \frac{14}{61}$, $\frac{1}{499712} a^{27} + \frac{7}{124928} a^{23} + \frac{15}{62464} a^{21} - \frac{3}{7808} a^{19} + \frac{15}{15616} a^{17} + \frac{21}{7808} a^{15} + \frac{3}{1952} a^{13} + \frac{7}{488} a^{11} + \frac{13}{488} a^{9} + \frac{1}{488} a^{7} - \frac{2}{61} a^{5} - \frac{21}{122} a^{3} - \frac{14}{61} a$, $\frac{1}{60964864} a^{28} - \frac{3}{7620608} a^{26} + \frac{7}{15241216} a^{24} - \frac{1289}{7620608} a^{22} + \frac{13}{952576} a^{20} - \frac{125}{238144} a^{18} - \frac{903}{476288} a^{16} + \frac{243}{238144} a^{14} + \frac{17}{238144} a^{12} + \frac{849}{119072} a^{10} - \frac{337}{14884} a^{8} - \frac{203}{29768} a^{6} + \frac{1539}{14884} a^{4} + \frac{424}{3721} a^{2} + \frac{534}{3721}$, $\frac{1}{60964864} a^{29} - \frac{3}{7620608} a^{27} + \frac{7}{15241216} a^{25} - \frac{1289}{7620608} a^{23} + \frac{13}{952576} a^{21} - \frac{125}{238144} a^{19} - \frac{903}{476288} a^{17} + \frac{243}{238144} a^{15} + \frac{17}{238144} a^{13} + \frac{849}{119072} a^{11} - \frac{337}{14884} a^{9} - \frac{203}{29768} a^{7} + \frac{1539}{14884} a^{5} + \frac{424}{3721} a^{3} + \frac{534}{3721} a$, $\frac{1}{201860478345873256042052882272583303882629634796322816} a^{30} + \frac{245123265655362253689065314151889497859696885}{50465119586468314010513220568145825970657408699080704} a^{28} - \frac{2112886561781305654042692496442881241285562323}{12616279896617078502628305142036456492664352174770176} a^{26} + \frac{14755960071945527389751995127952296686445231}{12993079193220472196321632484074620486781001209856} a^{24} + \frac{2680036503117950843139671830562881436293641684565}{12616279896617078502628305142036456492664352174770176} a^{22} - \frac{753358662599018459846842343717450244441471876847}{1577034987077134812828538142754557061583044021846272} a^{20} + \frac{823823569032494196415000723780134210412362658827}{3154069974154269625657076285509114123166088043692544} a^{18} + \frac{2164973509160085569455317119806940470692554741327}{1577034987077134812828538142754557061583044021846272} a^{16} + \frac{512021127111393137782122550056571876478175014117}{788517493538567406414269071377278530791522010923136} a^{14} + \frac{6805504712517969666490844342816895911708567327}{394258746769283703207134535688639265395761005461568} a^{12} - \frac{1145285774732348460786643458845171686684721081709}{197129373384641851603567267844319632697880502730784} a^{10} - \frac{1252467059091478869589761926725958528293858916435}{98564686692320925801783633922159816348940251365392} a^{8} + \frac{1024990450345270074233334148983731422466396304259}{49282343346160462900891816961079908174470125682696} a^{6} + \frac{2984633444043698815465661888051649950335157733603}{24641171673080231450445908480539954087235062841348} a^{4} - \frac{75788799137323362451775352002529622640338250893}{6160292918270057862611477120134988521808765710337} a^{2} - \frac{1392490906508925479664348349638778278128277265499}{6160292918270057862611477120134988521808765710337}$, $\frac{1}{201860478345873256042052882272583303882629634796322816} a^{31} + \frac{245123265655362253689065314151889497859696885}{50465119586468314010513220568145825970657408699080704} a^{29} - \frac{2112886561781305654042692496442881241285562323}{12616279896617078502628305142036456492664352174770176} a^{27} + \frac{14755960071945527389751995127952296686445231}{12993079193220472196321632484074620486781001209856} a^{25} + \frac{2680036503117950843139671830562881436293641684565}{12616279896617078502628305142036456492664352174770176} a^{23} - \frac{753358662599018459846842343717450244441471876847}{1577034987077134812828538142754557061583044021846272} a^{21} + \frac{823823569032494196415000723780134210412362658827}{3154069974154269625657076285509114123166088043692544} a^{19} + \frac{2164973509160085569455317119806940470692554741327}{1577034987077134812828538142754557061583044021846272} a^{17} + \frac{512021127111393137782122550056571876478175014117}{788517493538567406414269071377278530791522010923136} a^{15} + \frac{6805504712517969666490844342816895911708567327}{394258746769283703207134535688639265395761005461568} a^{13} - \frac{1145285774732348460786643458845171686684721081709}{197129373384641851603567267844319632697880502730784} a^{11} - \frac{1252467059091478869589761926725958528293858916435}{98564686692320925801783633922159816348940251365392} a^{9} + \frac{1024990450345270074233334148983731422466396304259}{49282343346160462900891816961079908174470125682696} a^{7} + \frac{2984633444043698815465661888051649950335157733603}{24641171673080231450445908480539954087235062841348} a^{5} - \frac{75788799137323362451775352002529622640338250893}{6160292918270057862611477120134988521808765710337} a^{3} - \frac{1392490906508925479664348349638778278128277265499}{6160292918270057862611477120134988521808765710337} a$

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:  $31$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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

$C_{32}$ (as 32T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 32
The 32 conjugacy class representatives for $C_{32}$
Character table for $C_{32}$ is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1, 16.16.633251189136789386043275954593.1

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 $16^{2}$ $32$ $32$ $16^{2}$ $32$ $32$ $32$ $32$ $32$ $16^{2}$ $32$ $32$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $16^{2}$ $32$

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
97Data not computed