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Properties

Label 32.0.368...256.1
Degree $32$
Signature $[0, 16]$
Discriminant $3.686\times 10^{50}$
Root discriminant $38.04$
Ramified primes $2, 41, 1277$
Class number $32$ (GRH)
Class group $[4, 8]$ (GRH)
Galois group 32T96908

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 6*x^28 + 49*x^24 - 248*x^20 + 1072*x^16 - 3968*x^12 + 12544*x^8 - 24576*x^4 + 65536)
 
gp: K = bnfinit(x^32 - 6*x^28 + 49*x^24 - 248*x^20 + 1072*x^16 - 3968*x^12 + 12544*x^8 - 24576*x^4 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, 0, 0, 0, -24576, 0, 0, 0, 12544, 0, 0, 0, -3968, 0, 0, 0, 1072, 0, 0, 0, -248, 0, 0, 0, 49, 0, 0, 0, -6, 0, 0, 0, 1]);
 

\( x^{32} - 6 x^{28} + 49 x^{24} - 248 x^{20} + 1072 x^{16} - 3968 x^{12} + 12544 x^{8} - 24576 x^{4} + 65536 \)

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:  \(368622913612148362854567389427658855145582683488256\)\(\medspace = 2^{64}\cdot 41^{4}\cdot 1277^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $38.04$
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:  $2, 41, 1277$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{4}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{3}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{10} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{11} + \frac{1}{8} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} - \frac{1}{16} a^{8} + \frac{3}{16} a^{6} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{16} a^{9} + \frac{3}{16} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{16} a^{14} - \frac{1}{8} a^{8} + \frac{3}{16} a^{6} - \frac{1}{8} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{16} - \frac{1}{16} a^{10} + \frac{3}{32} a^{8} - \frac{1}{16} a^{6} - \frac{1}{2} a^{5} + \frac{1}{8} a^{4}$, $\frac{1}{32} a^{17} - \frac{1}{16} a^{11} - \frac{1}{32} a^{9} + \frac{3}{16} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{18} - \frac{1}{32} a^{14} + \frac{1}{64} a^{10} - \frac{3}{16} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{128} a^{19} - \frac{1}{64} a^{15} - \frac{7}{128} a^{11} - \frac{5}{32} a^{7} - \frac{1}{2} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{256} a^{20} - \frac{1}{128} a^{16} - \frac{7}{256} a^{12} - \frac{5}{64} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{5}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{512} a^{21} + \frac{3}{256} a^{17} + \frac{9}{512} a^{13} - \frac{1}{16} a^{11} - \frac{3}{128} a^{9} + \frac{3}{16} a^{7} - \frac{11}{32} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{512} a^{22} - \frac{1}{256} a^{18} - \frac{7}{512} a^{14} + \frac{3}{128} a^{10} - \frac{1}{32} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{1024} a^{23} - \frac{1}{512} a^{19} - \frac{7}{1024} a^{15} - \frac{13}{256} a^{11} - \frac{5}{64} a^{7} - \frac{1}{2} a^{5} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{2048} a^{24} - \frac{1}{1024} a^{22} + \frac{1}{1024} a^{20} - \frac{3}{512} a^{18} + \frac{17}{2048} a^{16} + \frac{23}{1024} a^{14} - \frac{1}{128} a^{12} + \frac{11}{256} a^{10} + \frac{1}{16} a^{8} - \frac{5}{64} a^{6} - \frac{1}{2} a^{5} - \frac{1}{32} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4096} a^{25} - \frac{1}{2048} a^{23} + \frac{1}{2048} a^{21} - \frac{3}{1024} a^{19} + \frac{17}{4096} a^{17} + \frac{23}{2048} a^{15} + \frac{7}{256} a^{13} + \frac{27}{512} a^{11} - \frac{17}{128} a^{7} - \frac{5}{64} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4096} a^{26} - \frac{1}{2048} a^{22} - \frac{1}{512} a^{20} - \frac{7}{4096} a^{18} - \frac{3}{256} a^{16} - \frac{13}{1024} a^{14} - \frac{9}{512} a^{12} + \frac{11}{256} a^{10} + \frac{3}{128} a^{8} - \frac{3}{32} a^{6} - \frac{5}{32} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{8192} a^{27} - \frac{1}{4096} a^{23} - \frac{1}{1024} a^{21} - \frac{7}{8192} a^{19} + \frac{5}{512} a^{17} + \frac{51}{2048} a^{15} - \frac{9}{1024} a^{13} + \frac{27}{512} a^{11} - \frac{1}{256} a^{9} + \frac{5}{64} a^{7} - \frac{5}{64} a^{5} - \frac{1}{2} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{491520} a^{28} - \frac{11}{245760} a^{24} + \frac{401}{491520} a^{20} - \frac{1}{256} a^{19} - \frac{1}{64} a^{17} - \frac{833}{61440} a^{16} - \frac{3}{128} a^{15} - \frac{1}{32} a^{13} - \frac{949}{30720} a^{12} + \frac{7}{256} a^{11} + \frac{3}{64} a^{9} - \frac{53}{3840} a^{8} + \frac{7}{64} a^{7} + \frac{5}{16} a^{5} + \frac{953}{1920} a^{4} - \frac{5}{16} a^{3} - \frac{1}{4} a + \frac{1}{120}$, $\frac{1}{983040} a^{29} - \frac{11}{491520} a^{25} + \frac{401}{983040} a^{21} - \frac{1}{128} a^{18} - \frac{833}{122880} a^{17} + \frac{1}{64} a^{14} - \frac{949}{61440} a^{13} + \frac{7}{128} a^{10} - \frac{53}{7680} a^{9} - \frac{1}{4} a^{7} + \frac{5}{32} a^{6} + \frac{1913}{3840} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{3}{8} a^{2} - \frac{119}{240} a - \frac{1}{2}$, $\frac{1}{3932160} a^{30} - \frac{1}{983040} a^{28} - \frac{11}{1966080} a^{26} + \frac{11}{491520} a^{24} + \frac{401}{3932160} a^{22} - \frac{401}{983040} a^{20} - \frac{1}{256} a^{19} - \frac{833}{491520} a^{18} + \frac{833}{122880} a^{16} - \frac{3}{128} a^{15} + \frac{6731}{245760} a^{14} + \frac{949}{61440} a^{12} - \frac{9}{256} a^{11} + \frac{1867}{30720} a^{10} - \frac{907}{7680} a^{8} - \frac{13}{64} a^{7} + \frac{473}{15360} a^{6} - \frac{1}{2} a^{5} + \frac{1447}{3840} a^{4} - \frac{1}{16} a^{3} - \frac{479}{960} a^{2} - \frac{1}{240}$, $\frac{1}{7864320} a^{31} - \frac{1}{1966080} a^{29} - \frac{11}{3932160} a^{27} + \frac{11}{983040} a^{25} + \frac{401}{7864320} a^{23} - \frac{401}{1966080} a^{21} - \frac{833}{983040} a^{19} + \frac{833}{245760} a^{17} + \frac{6731}{491520} a^{15} + \frac{949}{122880} a^{13} - \frac{1973}{61440} a^{11} - \frac{907}{15360} a^{9} - \frac{1}{8} a^{8} + \frac{6233}{30720} a^{7} - \frac{1}{4} a^{6} - \frac{473}{7680} a^{5} + \frac{3}{8} a^{4} - \frac{479}{1920} a^{3} + \frac{1}{4} a^{2} + \frac{239}{480} a - \frac{1}{2}$

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

Class group and class number

$C_{4}\times C_{8}$, which has order $32$ (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:  \( -\frac{23}{1966080} a^{31} + \frac{13}{983040} a^{27} - \frac{583}{1966080} a^{23} + \frac{379}{245760} a^{19} - \frac{1093}{122880} a^{15} + \frac{199}{15360} a^{11} - \frac{319}{7680} a^{7} + \frac{37}{480} a^{3} \) (order $8$)
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:  \( 747032345387.1615 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 747032345387.1615 \cdot 32}{8\sqrt{368622913612148362854567389427658855145582683488256}}\approx 0.918303263736995$ (assuming GRH)

Galois group

32T96908:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 1536
The 80 conjugacy class representatives for t32n96908 are not computed
Character table for t32n96908 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), 4.4.40864.1, \(\Q(\zeta_{8})\), 8.8.106871455744.1, Deg 8, 8.0.26717863936.1, Deg 8, 8.0.68464526336.1, 8.0.6679465984.1, 8.8.273858105344.1, Deg 16, Deg 16, 16.0.19199555036826982747734016.1, 16.0.1199972189801686421733376.1, Deg 16, Deg 16, 16.0.74998261862605401358336.1

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

Sibling fields

Degree 32 siblings: data not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{8}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
1277Data not computed