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Properties

Label 32.32.1670642877...5088.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{32}\cdot 97^{31}$
Root discriminant $168.16$
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![97, 0, -25026, 0, 1096973, 0, -18792198, 0, 148892090, 0, -561198156, 0, 1039362469, 0, -1069475149, 0, 665320090, 0, -262762815, 0, 67589988, 0, -11436203, 0, 1268954, 0, -90598, 0, 3977, 0, -97, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 97*x^30 + 3977*x^28 - 90598*x^26 + 1268954*x^24 - 11436203*x^22 + 67589988*x^20 - 262762815*x^18 + 665320090*x^16 - 1069475149*x^14 + 1039362469*x^12 - 561198156*x^10 + 148892090*x^8 - 18792198*x^6 + 1096973*x^4 - 25026*x^2 + 97)
 
gp: K = bnfinit(x^32 - 97*x^30 + 3977*x^28 - 90598*x^26 + 1268954*x^24 - 11436203*x^22 + 67589988*x^20 - 262762815*x^18 + 665320090*x^16 - 1069475149*x^14 + 1039362469*x^12 - 561198156*x^10 + 148892090*x^8 - 18792198*x^6 + 1096973*x^4 - 25026*x^2 + 97, 1)
 

Normalized defining polynomial

\( x^{32} - 97 x^{30} + 3977 x^{28} - 90598 x^{26} + 1268954 x^{24} - 11436203 x^{22} + 67589988 x^{20} - 262762815 x^{18} + 665320090 x^{16} - 1069475149 x^{14} + 1039362469 x^{12} - 561198156 x^{10} + 148892090 x^{8} - 18792198 x^{6} + 1096973 x^{4} - 25026 x^{2} + 97 \)

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:  \(167064287751196233763024674646775194239336663364043996417475105771225088=2^{32}\cdot 97^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $168.16$
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:  \(388=2^{2}\cdot 97\)
Dirichlet character group:    $\lbrace$$\chi_{388}(1,·)$, $\chi_{388}(131,·)$, $\chi_{388}(263,·)$, $\chi_{388}(139,·)$, $\chi_{388}(269,·)$, $\chi_{388}(143,·)$, $\chi_{388}(273,·)$, $\chi_{388}(19,·)$, $\chi_{388}(33,·)$, $\chi_{388}(175,·)$, $\chi_{388}(51,·)$, $\chi_{388}(309,·)$, $\chi_{388}(55,·)$, $\chi_{388}(313,·)$, $\chi_{388}(63,·)$, $\chi_{388}(193,·)$, $\chi_{388}(67,·)$, $\chi_{388}(161,·)$, $\chi_{388}(311,·)$, $\chi_{388}(341,·)$, $\chi_{388}(343,·)$, $\chi_{388}(89,·)$, $\chi_{388}(271,·)$, $\chi_{388}(221,·)$, $\chi_{388}(105,·)$, $\chi_{388}(127,·)$, $\chi_{388}(109,·)$, $\chi_{388}(239,·)$, $\chi_{388}(241,·)$, $\chi_{388}(361,·)$, $\chi_{388}(319,·)$, $\chi_{388}(85,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{61} a^{24} + \frac{15}{61} a^{22} - \frac{12}{61} a^{20} + \frac{18}{61} a^{18} + \frac{14}{61} a^{16} - \frac{19}{61} a^{14} - \frac{20}{61} a^{12} + \frac{11}{61} a^{10} + \frac{10}{61} a^{8} - \frac{7}{61} a^{6} + \frac{18}{61} a^{4} + \frac{18}{61} a^{2} + \frac{5}{61}$, $\frac{1}{61} a^{25} + \frac{15}{61} a^{23} - \frac{12}{61} a^{21} + \frac{18}{61} a^{19} + \frac{14}{61} a^{17} - \frac{19}{61} a^{15} - \frac{20}{61} a^{13} + \frac{11}{61} a^{11} + \frac{10}{61} a^{9} - \frac{7}{61} a^{7} + \frac{18}{61} a^{5} + \frac{18}{61} a^{3} + \frac{5}{61} a$, $\frac{1}{61} a^{26} + \frac{7}{61} a^{22} + \frac{15}{61} a^{20} - \frac{12}{61} a^{18} + \frac{15}{61} a^{16} + \frac{21}{61} a^{14} + \frac{6}{61} a^{12} + \frac{28}{61} a^{10} + \frac{26}{61} a^{8} + \frac{1}{61} a^{6} - \frac{8}{61} a^{4} - \frac{21}{61} a^{2} - \frac{14}{61}$, $\frac{1}{61} a^{27} + \frac{7}{61} a^{23} + \frac{15}{61} a^{21} - \frac{12}{61} a^{19} + \frac{15}{61} a^{17} + \frac{21}{61} a^{15} + \frac{6}{61} a^{13} + \frac{28}{61} a^{11} + \frac{26}{61} a^{9} + \frac{1}{61} a^{7} - \frac{8}{61} a^{5} - \frac{21}{61} a^{3} - \frac{14}{61} a$, $\frac{1}{3721} a^{28} - \frac{12}{3721} a^{26} + \frac{7}{3721} a^{24} - \frac{1289}{3721} a^{22} + \frac{52}{3721} a^{20} - \frac{1000}{3721} a^{18} - \frac{1806}{3721} a^{16} + \frac{486}{3721} a^{14} + \frac{17}{3721} a^{12} + \frac{849}{3721} a^{10} - \frac{1348}{3721} a^{8} - \frac{203}{3721} a^{6} + \frac{1539}{3721} a^{4} + \frac{848}{3721} a^{2} + \frac{534}{3721}$, $\frac{1}{3721} a^{29} - \frac{12}{3721} a^{27} + \frac{7}{3721} a^{25} - \frac{1289}{3721} a^{23} + \frac{52}{3721} a^{21} - \frac{1000}{3721} a^{19} - \frac{1806}{3721} a^{17} + \frac{486}{3721} a^{15} + \frac{17}{3721} a^{13} + \frac{849}{3721} a^{11} - \frac{1348}{3721} a^{9} - \frac{203}{3721} a^{7} + \frac{1539}{3721} a^{5} + \frac{848}{3721} a^{3} + \frac{534}{3721} a$, $\frac{1}{6160292918270057862611477120134988521808765710337} a^{30} + \frac{490246531310724507378130628303778995719393770}{6160292918270057862611477120134988521808765710337} a^{28} - \frac{8451546247125222616170769985771524965142249292}{6160292918270057862611477120134988521808765710337} a^{26} + \frac{29511920143891054779503990255904593372890462}{6344276949814683689610172111364560784561035747} a^{24} + \frac{2680036503117950843139671830562881436293641684565}{6160292918270057862611477120134988521808765710337} a^{22} - \frac{3013434650396073839387369374869800977765887507388}{6160292918270057862611477120134988521808765710337} a^{20} + \frac{823823569032494196415000723780134210412362658827}{6160292918270057862611477120134988521808765710337} a^{18} + \frac{2164973509160085569455317119806940470692554741327}{6160292918270057862611477120134988521808765710337} a^{16} + \frac{512021127111393137782122550056571876478175014117}{6160292918270057862611477120134988521808765710337} a^{14} + \frac{6805504712517969666490844342816895911708567327}{6160292918270057862611477120134988521808765710337} a^{12} - \frac{1145285774732348460786643458845171686684721081709}{6160292918270057862611477120134988521808765710337} a^{10} - \frac{1252467059091478869589761926725958528293858916435}{6160292918270057862611477120134988521808765710337} a^{8} + \frac{1024990450345270074233334148983731422466396304259}{6160292918270057862611477120134988521808765710337} a^{6} + \frac{2984633444043698815465661888051649950335157733603}{6160292918270057862611477120134988521808765710337} a^{4} - \frac{151577598274646724903550704005059245280676501786}{6160292918270057862611477120134988521808765710337} a^{2} - \frac{1392490906508925479664348349638778278128277265499}{6160292918270057862611477120134988521808765710337}$, $\frac{1}{6160292918270057862611477120134988521808765710337} a^{31} + \frac{490246531310724507378130628303778995719393770}{6160292918270057862611477120134988521808765710337} a^{29} - \frac{8451546247125222616170769985771524965142249292}{6160292918270057862611477120134988521808765710337} a^{27} + \frac{29511920143891054779503990255904593372890462}{6344276949814683689610172111364560784561035747} a^{25} + \frac{2680036503117950843139671830562881436293641684565}{6160292918270057862611477120134988521808765710337} a^{23} - \frac{3013434650396073839387369374869800977765887507388}{6160292918270057862611477120134988521808765710337} a^{21} + \frac{823823569032494196415000723780134210412362658827}{6160292918270057862611477120134988521808765710337} a^{19} + \frac{2164973509160085569455317119806940470692554741327}{6160292918270057862611477120134988521808765710337} a^{17} + \frac{512021127111393137782122550056571876478175014117}{6160292918270057862611477120134988521808765710337} a^{15} + \frac{6805504712517969666490844342816895911708567327}{6160292918270057862611477120134988521808765710337} a^{13} - \frac{1145285774732348460786643458845171686684721081709}{6160292918270057862611477120134988521808765710337} a^{11} - \frac{1252467059091478869589761926725958528293858916435}{6160292918270057862611477120134988521808765710337} a^{9} + \frac{1024990450345270074233334148983731422466396304259}{6160292918270057862611477120134988521808765710337} a^{7} + \frac{2984633444043698815465661888051649950335157733603}{6160292918270057862611477120134988521808765710337} a^{5} - \frac{151577598274646724903550704005059245280676501786}{6160292918270057862611477120134988521808765710337} a^{3} - \frac{1392490906508925479664348349638778278128277265499}{6160292918270057862611477120134988521808765710337} a$

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