# Properties

 Label 32.0.10948725162...7168.1 Degree $32$ Signature $[0, 16]$ 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)

# Related objects

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)

## Normalizeddefining 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: $[0, 16]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$10948725162062396375893585077651059129669167570225987349215648531823007367168=2^{48}\cdot 97^{31}$$ magma: Discriminant(K);  sage: K.disc()  gp: K.disc Root discriminant: $237.81$ magma: Abs(Discriminant(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(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$$\chi_{776}(1,·), \chi_{776}(645,·), \chi_{776}(273,·), \chi_{776}(149,·), \chi_{776}(473,·), \chi_{776}(261,·), \chi_{776}(33,·), \chi_{776}(325,·), \chi_{776}(241,·), \chi_{776}(45,·), \chi_{776}(709,·), \chi_{776}(697,·), \chi_{776}(117,·), \chi_{776}(757,·), \chi_{776}(193,·), \chi_{776}(69,·), \chi_{776}(161,·), \chi_{776}(77,·), \chi_{776}(333,·), \chi_{776}(725,·), \chi_{776}(313,·), \chi_{776}(729,·), \chi_{776}(609,·), \chi_{776}(657,·), \chi_{776}(105,·), \chi_{776}(637,·), \chi_{776}(497,·), \chi_{776}(245,·), \chi_{776}(89,·), \chi_{776}(361,·), \chi_{776}(125,·), \chi_{776}(213,·)$$\rbrace$ This is a CM field.

## Integral basis (with respect to field generator $$a$$)

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

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: $15$ 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

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 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