Properties

Label 32.0.272...249.1
Degree $32$
Signature $[0, 16]$
Discriminant $2.723\times 10^{50}$
Root discriminant $37.68$
Ramified primes $7, 17$
Class number $45$ (GRH)
Class group $[3, 15]$ (GRH)
Galois group $C_2\times C_{16}$ (as 32T32)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 - x^30 + 3*x^29 - x^28 - 5*x^27 + 7*x^26 + 3*x^25 - 17*x^24 + 11*x^23 + 23*x^22 - 45*x^21 - x^20 + 91*x^19 - 89*x^18 - 93*x^17 + 271*x^16 - 186*x^15 - 356*x^14 + 728*x^13 - 16*x^12 - 1440*x^11 + 1472*x^10 + 1408*x^9 - 4352*x^8 + 1536*x^7 + 7168*x^6 - 10240*x^5 - 4096*x^4 + 24576*x^3 - 16384*x^2 - 32768*x + 65536)
 
gp: K = bnfinit(x^32 - x^31 - x^30 + 3*x^29 - x^28 - 5*x^27 + 7*x^26 + 3*x^25 - 17*x^24 + 11*x^23 + 23*x^22 - 45*x^21 - x^20 + 91*x^19 - 89*x^18 - 93*x^17 + 271*x^16 - 186*x^15 - 356*x^14 + 728*x^13 - 16*x^12 - 1440*x^11 + 1472*x^10 + 1408*x^9 - 4352*x^8 + 1536*x^7 + 7168*x^6 - 10240*x^5 - 4096*x^4 + 24576*x^3 - 16384*x^2 - 32768*x + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, -32768, -16384, 24576, -4096, -10240, 7168, 1536, -4352, 1408, 1472, -1440, -16, 728, -356, -186, 271, -93, -89, 91, -1, -45, 23, 11, -17, 3, 7, -5, -1, 3, -1, -1, 1]);
 

\( x^{32} - x^{31} - x^{30} + 3 x^{29} - x^{28} - 5 x^{27} + 7 x^{26} + 3 x^{25} - 17 x^{24} + 11 x^{23} + 23 x^{22} - 45 x^{21} - x^{20} + 91 x^{19} - 89 x^{18} - 93 x^{17} + 271 x^{16} - 186 x^{15} - 356 x^{14} + 728 x^{13} - 16 x^{12} - 1440 x^{11} + 1472 x^{10} + 1408 x^{9} - 4352 x^{8} + 1536 x^{7} + 7168 x^{6} - 10240 x^{5} - 4096 x^{4} + 24576 x^{3} - 16384 x^{2} - 32768 x + 65536 \)

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

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(272292877590407567597186433188495333554574218609249\)\(\medspace = 7^{16}\cdot 17^{30}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $37.68$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $7, 17$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $32$
This field is Galois and abelian over $\Q$.
Conductor:  \(119=7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{119}(1,·)$, $\chi_{119}(6,·)$, $\chi_{119}(8,·)$, $\chi_{119}(13,·)$, $\chi_{119}(15,·)$, $\chi_{119}(20,·)$, $\chi_{119}(22,·)$, $\chi_{119}(27,·)$, $\chi_{119}(29,·)$, $\chi_{119}(36,·)$, $\chi_{119}(41,·)$, $\chi_{119}(43,·)$, $\chi_{119}(48,·)$, $\chi_{119}(50,·)$, $\chi_{119}(55,·)$, $\chi_{119}(57,·)$, $\chi_{119}(62,·)$, $\chi_{119}(64,·)$, $\chi_{119}(69,·)$, $\chi_{119}(71,·)$, $\chi_{119}(76,·)$, $\chi_{119}(78,·)$, $\chi_{119}(83,·)$, $\chi_{119}(90,·)$, $\chi_{119}(92,·)$, $\chi_{119}(97,·)$, $\chi_{119}(99,·)$, $\chi_{119}(104,·)$, $\chi_{119}(106,·)$, $\chi_{119}(111,·)$, $\chi_{119}(113,·)$, $\chi_{119}(118,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{542} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{93}{271}$, $\frac{1}{1084} a^{18} - \frac{1}{1084} a^{17} + \frac{1}{4} a^{16} + \frac{1}{4} a^{15} + \frac{1}{4} a^{14} + \frac{1}{4} a^{13} + \frac{1}{4} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{93}{542} a - \frac{89}{271}$, $\frac{1}{2168} a^{19} - \frac{1}{2168} a^{18} - \frac{1}{2168} a^{17} - \frac{3}{8} a^{16} + \frac{1}{8} a^{15} - \frac{3}{8} a^{14} + \frac{1}{8} a^{13} - \frac{3}{8} a^{12} + \frac{1}{8} a^{11} - \frac{3}{8} a^{10} + \frac{1}{8} a^{9} - \frac{3}{8} a^{8} + \frac{1}{8} a^{7} - \frac{3}{8} a^{6} + \frac{1}{8} a^{5} - \frac{3}{8} a^{4} + \frac{1}{8} a^{3} - \frac{93}{1084} a^{2} - \frac{89}{542} a + \frac{91}{271}$, $\frac{1}{4336} a^{20} - \frac{1}{4336} a^{19} - \frac{1}{4336} a^{18} + \frac{3}{4336} a^{17} + \frac{1}{16} a^{16} + \frac{5}{16} a^{15} - \frac{7}{16} a^{14} - \frac{3}{16} a^{13} + \frac{1}{16} a^{12} + \frac{5}{16} a^{11} - \frac{7}{16} a^{10} - \frac{3}{16} a^{9} + \frac{1}{16} a^{8} + \frac{5}{16} a^{7} - \frac{7}{16} a^{6} - \frac{3}{16} a^{5} + \frac{1}{16} a^{4} - \frac{93}{2168} a^{3} - \frac{89}{1084} a^{2} + \frac{91}{542} a - \frac{1}{271}$, $\frac{1}{8672} a^{21} - \frac{1}{8672} a^{20} - \frac{1}{8672} a^{19} + \frac{3}{8672} a^{18} - \frac{1}{8672} a^{17} - \frac{11}{32} a^{16} + \frac{9}{32} a^{15} + \frac{13}{32} a^{14} + \frac{1}{32} a^{13} + \frac{5}{32} a^{12} - \frac{7}{32} a^{11} - \frac{3}{32} a^{10} - \frac{15}{32} a^{9} - \frac{11}{32} a^{8} + \frac{9}{32} a^{7} + \frac{13}{32} a^{6} + \frac{1}{32} a^{5} - \frac{93}{4336} a^{4} - \frac{89}{2168} a^{3} + \frac{91}{1084} a^{2} - \frac{1}{542} a - \frac{45}{271}$, $\frac{1}{17344} a^{22} - \frac{1}{17344} a^{21} - \frac{1}{17344} a^{20} + \frac{3}{17344} a^{19} - \frac{1}{17344} a^{18} - \frac{5}{17344} a^{17} + \frac{9}{64} a^{16} + \frac{13}{64} a^{15} - \frac{31}{64} a^{14} + \frac{5}{64} a^{13} - \frac{7}{64} a^{12} - \frac{3}{64} a^{11} + \frac{17}{64} a^{10} - \frac{11}{64} a^{9} - \frac{23}{64} a^{8} - \frac{19}{64} a^{7} + \frac{1}{64} a^{6} - \frac{93}{8672} a^{5} - \frac{89}{4336} a^{4} + \frac{91}{2168} a^{3} - \frac{1}{1084} a^{2} - \frac{45}{542} a + \frac{23}{271}$, $\frac{1}{34688} a^{23} - \frac{1}{34688} a^{22} - \frac{1}{34688} a^{21} + \frac{3}{34688} a^{20} - \frac{1}{34688} a^{19} - \frac{5}{34688} a^{18} + \frac{7}{34688} a^{17} - \frac{51}{128} a^{16} + \frac{33}{128} a^{15} - \frac{59}{128} a^{14} - \frac{7}{128} a^{13} - \frac{3}{128} a^{12} + \frac{17}{128} a^{11} - \frac{11}{128} a^{10} - \frac{23}{128} a^{9} + \frac{45}{128} a^{8} + \frac{1}{128} a^{7} - \frac{93}{17344} a^{6} - \frac{89}{8672} a^{5} + \frac{91}{4336} a^{4} - \frac{1}{2168} a^{3} - \frac{45}{1084} a^{2} + \frac{23}{542} a + \frac{11}{271}$, $\frac{1}{69376} a^{24} - \frac{1}{69376} a^{23} - \frac{1}{69376} a^{22} + \frac{3}{69376} a^{21} - \frac{1}{69376} a^{20} - \frac{5}{69376} a^{19} + \frac{7}{69376} a^{18} + \frac{3}{69376} a^{17} + \frac{33}{256} a^{16} + \frac{69}{256} a^{15} + \frac{121}{256} a^{14} - \frac{3}{256} a^{13} + \frac{17}{256} a^{12} - \frac{11}{256} a^{11} - \frac{23}{256} a^{10} + \frac{45}{256} a^{9} + \frac{1}{256} a^{8} - \frac{93}{34688} a^{7} - \frac{89}{17344} a^{6} + \frac{91}{8672} a^{5} - \frac{1}{4336} a^{4} - \frac{45}{2168} a^{3} + \frac{23}{1084} a^{2} + \frac{11}{542} a - \frac{17}{271}$, $\frac{1}{138752} a^{25} - \frac{1}{138752} a^{24} - \frac{1}{138752} a^{23} + \frac{3}{138752} a^{22} - \frac{1}{138752} a^{21} - \frac{5}{138752} a^{20} + \frac{7}{138752} a^{19} + \frac{3}{138752} a^{18} - \frac{17}{138752} a^{17} - \frac{187}{512} a^{16} + \frac{121}{512} a^{15} + \frac{253}{512} a^{14} + \frac{17}{512} a^{13} - \frac{11}{512} a^{12} - \frac{23}{512} a^{11} + \frac{45}{512} a^{10} + \frac{1}{512} a^{9} - \frac{93}{69376} a^{8} - \frac{89}{34688} a^{7} + \frac{91}{17344} a^{6} - \frac{1}{8672} a^{5} - \frac{45}{4336} a^{4} + \frac{23}{2168} a^{3} + \frac{11}{1084} a^{2} - \frac{17}{542} a + \frac{3}{271}$, $\frac{1}{277504} a^{26} - \frac{1}{277504} a^{25} - \frac{1}{277504} a^{24} + \frac{3}{277504} a^{23} - \frac{1}{277504} a^{22} - \frac{5}{277504} a^{21} + \frac{7}{277504} a^{20} + \frac{3}{277504} a^{19} - \frac{17}{277504} a^{18} + \frac{11}{277504} a^{17} + \frac{121}{1024} a^{16} + \frac{253}{1024} a^{15} - \frac{495}{1024} a^{14} - \frac{11}{1024} a^{13} - \frac{23}{1024} a^{12} + \frac{45}{1024} a^{11} + \frac{1}{1024} a^{10} - \frac{93}{138752} a^{9} - \frac{89}{69376} a^{8} + \frac{91}{34688} a^{7} - \frac{1}{17344} a^{6} - \frac{45}{8672} a^{5} + \frac{23}{4336} a^{4} + \frac{11}{2168} a^{3} - \frac{17}{1084} a^{2} + \frac{3}{542} a + \frac{7}{271}$, $\frac{1}{555008} a^{27} - \frac{1}{555008} a^{26} - \frac{1}{555008} a^{25} + \frac{3}{555008} a^{24} - \frac{1}{555008} a^{23} - \frac{5}{555008} a^{22} + \frac{7}{555008} a^{21} + \frac{3}{555008} a^{20} - \frac{17}{555008} a^{19} + \frac{11}{555008} a^{18} + \frac{23}{555008} a^{17} - \frac{771}{2048} a^{16} + \frac{529}{2048} a^{15} + \frac{1013}{2048} a^{14} - \frac{23}{2048} a^{13} + \frac{45}{2048} a^{12} + \frac{1}{2048} a^{11} - \frac{93}{277504} a^{10} - \frac{89}{138752} a^{9} + \frac{91}{69376} a^{8} - \frac{1}{34688} a^{7} - \frac{45}{17344} a^{6} + \frac{23}{8672} a^{5} + \frac{11}{4336} a^{4} - \frac{17}{2168} a^{3} + \frac{3}{1084} a^{2} + \frac{7}{542} a - \frac{5}{271}$, $\frac{1}{1110016} a^{28} - \frac{1}{1110016} a^{27} - \frac{1}{1110016} a^{26} + \frac{3}{1110016} a^{25} - \frac{1}{1110016} a^{24} - \frac{5}{1110016} a^{23} + \frac{7}{1110016} a^{22} + \frac{3}{1110016} a^{21} - \frac{17}{1110016} a^{20} + \frac{11}{1110016} a^{19} + \frac{23}{1110016} a^{18} - \frac{45}{1110016} a^{17} + \frac{529}{4096} a^{16} + \frac{1013}{4096} a^{15} + \frac{2025}{4096} a^{14} + \frac{45}{4096} a^{13} + \frac{1}{4096} a^{12} - \frac{93}{555008} a^{11} - \frac{89}{277504} a^{10} + \frac{91}{138752} a^{9} - \frac{1}{69376} a^{8} - \frac{45}{34688} a^{7} + \frac{23}{17344} a^{6} + \frac{11}{8672} a^{5} - \frac{17}{4336} a^{4} + \frac{3}{2168} a^{3} + \frac{7}{1084} a^{2} - \frac{5}{542} a - \frac{1}{271}$, $\frac{1}{2220032} a^{29} - \frac{1}{2220032} a^{28} - \frac{1}{2220032} a^{27} + \frac{3}{2220032} a^{26} - \frac{1}{2220032} a^{25} - \frac{5}{2220032} a^{24} + \frac{7}{2220032} a^{23} + \frac{3}{2220032} a^{22} - \frac{17}{2220032} a^{21} + \frac{11}{2220032} a^{20} + \frac{23}{2220032} a^{19} - \frac{45}{2220032} a^{18} - \frac{1}{2220032} a^{17} - \frac{3083}{8192} a^{16} + \frac{2025}{8192} a^{15} - \frac{4051}{8192} a^{14} + \frac{1}{8192} a^{13} - \frac{93}{1110016} a^{12} - \frac{89}{555008} a^{11} + \frac{91}{277504} a^{10} - \frac{1}{138752} a^{9} - \frac{45}{69376} a^{8} + \frac{23}{34688} a^{7} + \frac{11}{17344} a^{6} - \frac{17}{8672} a^{5} + \frac{3}{4336} a^{4} + \frac{7}{2168} a^{3} - \frac{5}{1084} a^{2} - \frac{1}{542} a + \frac{3}{271}$, $\frac{1}{4440064} a^{30} - \frac{1}{4440064} a^{29} - \frac{1}{4440064} a^{28} + \frac{3}{4440064} a^{27} - \frac{1}{4440064} a^{26} - \frac{5}{4440064} a^{25} + \frac{7}{4440064} a^{24} + \frac{3}{4440064} a^{23} - \frac{17}{4440064} a^{22} + \frac{11}{4440064} a^{21} + \frac{23}{4440064} a^{20} - \frac{45}{4440064} a^{19} - \frac{1}{4440064} a^{18} + \frac{91}{4440064} a^{17} - \frac{6167}{16384} a^{16} - \frac{4051}{16384} a^{15} + \frac{1}{16384} a^{14} - \frac{93}{2220032} a^{13} - \frac{89}{1110016} a^{12} + \frac{91}{555008} a^{11} - \frac{1}{277504} a^{10} - \frac{45}{138752} a^{9} + \frac{23}{69376} a^{8} + \frac{11}{34688} a^{7} - \frac{17}{17344} a^{6} + \frac{3}{8672} a^{5} + \frac{7}{4336} a^{4} - \frac{5}{2168} a^{3} - \frac{1}{1084} a^{2} + \frac{3}{542} a - \frac{1}{271}$, $\frac{1}{8880128} a^{31} - \frac{1}{8880128} a^{30} - \frac{1}{8880128} a^{29} + \frac{3}{8880128} a^{28} - \frac{1}{8880128} a^{27} - \frac{5}{8880128} a^{26} + \frac{7}{8880128} a^{25} + \frac{3}{8880128} a^{24} - \frac{17}{8880128} a^{23} + \frac{11}{8880128} a^{22} + \frac{23}{8880128} a^{21} - \frac{45}{8880128} a^{20} - \frac{1}{8880128} a^{19} + \frac{91}{8880128} a^{18} - \frac{89}{8880128} a^{17} + \frac{12333}{32768} a^{16} + \frac{1}{32768} a^{15} - \frac{93}{4440064} a^{14} - \frac{89}{2220032} a^{13} + \frac{91}{1110016} a^{12} - \frac{1}{555008} a^{11} - \frac{45}{277504} a^{10} + \frac{23}{138752} a^{9} + \frac{11}{69376} a^{8} - \frac{17}{34688} a^{7} + \frac{3}{17344} a^{6} + \frac{7}{8672} a^{5} - \frac{5}{4336} a^{4} - \frac{1}{2168} a^{3} + \frac{3}{1084} a^{2} - \frac{1}{542} a - \frac{1}{271}$

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

Class group and class number

$C_{3}\times C_{15}$, which has order $45$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{45}{1110016} a^{29} + \frac{8641}{1110016} a^{12} \) (order $34$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 308154877506.37555 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 308154877506.37555 \cdot 45}{34\sqrt{272292877590407567597186433188495333554574218609249}}\approx 0.145835211511193$ (assuming GRH)

Galois group

$C_2\times C_{16}$ (as 32T32):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_{16}$
Character table for $C_2\times C_{16}$ is not computed

Intermediate fields

\(\Q(\sqrt{-119}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-7}, \sqrt{17})\), 4.4.4913.1, 4.0.240737.1, 8.0.57954303169.1, \(\Q(\zeta_{17})^+\), 8.0.985223153873.1, 16.0.970664662927461034900129.1, 16.16.16501299269766837593302193.1, \(\Q(\zeta_{17})\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ R $16^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
17Data not computed