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Properties

Label 32.0.237...896.1
Degree $32$
Signature $[0, 16]$
Discriminant $2.379\times 10^{46}$
Root discriminant $28.14$
Ramified primes $2, 3, 17$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $C_2^2\times C_2^2:C_4$ (as 32T262)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^30 + 32*x^28 - 80*x^26 + 127*x^24 - 80*x^22 - 224*x^20 + 936*x^18 - 2175*x^16 + 3744*x^14 - 3584*x^12 - 5120*x^10 + 32512*x^8 - 81920*x^6 + 131072*x^4 - 131072*x^2 + 65536)
 
gp: K = bnfinit(x^32 - 8*x^30 + 32*x^28 - 80*x^26 + 127*x^24 - 80*x^22 - 224*x^20 + 936*x^18 - 2175*x^16 + 3744*x^14 - 3584*x^12 - 5120*x^10 + 32512*x^8 - 81920*x^6 + 131072*x^4 - 131072*x^2 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, 0, -131072, 0, 131072, 0, -81920, 0, 32512, 0, -5120, 0, -3584, 0, 3744, 0, -2175, 0, 936, 0, -224, 0, -80, 0, 127, 0, -80, 0, 32, 0, -8, 0, 1]);
 

\( x^{32} - 8 x^{30} + 32 x^{28} - 80 x^{26} + 127 x^{24} - 80 x^{22} - 224 x^{20} + 936 x^{18} - 2175 x^{16} + 3744 x^{14} - 3584 x^{12} - 5120 x^{10} + 32512 x^{8} - 81920 x^{6} + 131072 x^{4} - 131072 x^{2} + 65536 \)

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:  \(23790908696561643372461609312578223409406672896\)\(\medspace = 2^{96}\cdot 3^{16}\cdot 17^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $28.14$
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:  $2, 3, 17$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $16$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{10} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{19} - \frac{1}{8} a^{11} + \frac{1}{8} a^{3}$, $\frac{1}{112} a^{20} - \frac{1}{14} a^{18} + \frac{2}{7} a^{16} + \frac{47}{112} a^{12} + \frac{1}{7} a^{10} + \frac{3}{7} a^{8} - \frac{1}{2} a^{6} - \frac{31}{112} a^{4} - \frac{2}{7} a^{2} + \frac{1}{7}$, $\frac{1}{224} a^{21} - \frac{1}{28} a^{19} + \frac{1}{7} a^{17} - \frac{1}{2} a^{15} - \frac{65}{224} a^{13} + \frac{1}{14} a^{11} - \frac{2}{7} a^{9} - \frac{1}{4} a^{7} - \frac{31}{224} a^{5} - \frac{1}{7} a^{3} - \frac{3}{7} a$, $\frac{1}{448} a^{22} - \frac{1}{14} a^{18} - \frac{5}{28} a^{16} - \frac{65}{448} a^{14} + \frac{3}{8} a^{12} - \frac{5}{14} a^{10} + \frac{13}{56} a^{8} + \frac{193}{448} a^{6} + \frac{3}{8} a^{4} - \frac{2}{7} a^{2} + \frac{2}{7}$, $\frac{1}{896} a^{23} - \frac{1}{28} a^{19} - \frac{5}{56} a^{17} + \frac{383}{896} a^{15} + \frac{3}{16} a^{13} + \frac{9}{28} a^{11} - \frac{43}{112} a^{9} + \frac{193}{896} a^{7} + \frac{3}{16} a^{5} - \frac{1}{7} a^{3} + \frac{1}{7} a$, $\frac{1}{23296} a^{24} - \frac{19}{208} a^{18} + \frac{45}{256} a^{16} + \frac{15}{32} a^{14} + \frac{5}{26} a^{12} + \frac{15}{32} a^{10} + \frac{19}{256} a^{8} - \frac{157}{416} a^{6} + \frac{3}{8} a^{4} + \frac{16}{91}$, $\frac{1}{46592} a^{25} - \frac{19}{416} a^{19} + \frac{45}{512} a^{17} - \frac{17}{64} a^{15} - \frac{21}{52} a^{13} + \frac{15}{64} a^{11} + \frac{19}{512} a^{9} - \frac{157}{832} a^{7} - \frac{5}{16} a^{5} - \frac{75}{182} a$, $\frac{1}{93184} a^{26} + \frac{23}{5824} a^{20} + \frac{571}{7168} a^{18} + \frac{201}{896} a^{16} + \frac{31}{104} a^{14} + \frac{337}{896} a^{12} + \frac{1413}{7168} a^{10} - \frac{3595}{11648} a^{8} + \frac{11}{32} a^{6} + \frac{19}{112} a^{4} + \frac{17}{91} a^{2} + \frac{3}{7}$, $\frac{1}{186368} a^{27} + \frac{23}{11648} a^{21} + \frac{571}{14336} a^{19} + \frac{201}{1792} a^{17} - \frac{73}{208} a^{15} + \frac{337}{1792} a^{13} - \frac{5755}{14336} a^{11} - \frac{3595}{23296} a^{9} + \frac{11}{64} a^{7} - \frac{93}{224} a^{5} - \frac{37}{91} a^{3} - \frac{2}{7} a$, $\frac{1}{11554816} a^{28} + \frac{1}{222208} a^{26} + \frac{3}{722176} a^{24} + \frac{751}{722176} a^{22} + \frac{1275}{888832} a^{20} + \frac{2973}{2888704} a^{18} - \frac{71961}{722176} a^{16} + \frac{35281}{111104} a^{14} + \frac{1773025}{11554816} a^{12} + \frac{1161367}{2888704} a^{10} + \frac{14803}{55552} a^{8} + \frac{30053}{180544} a^{6} + \frac{1243}{22568} a^{4} - \frac{27}{217} a^{2} + \frac{1244}{2821}$, $\frac{1}{23109632} a^{29} + \frac{1}{444416} a^{27} + \frac{3}{1444352} a^{25} + \frac{751}{1444352} a^{23} + \frac{1275}{1777664} a^{21} + \frac{2973}{5777408} a^{19} - \frac{71961}{1444352} a^{17} - \frac{75823}{222208} a^{15} + \frac{1773025}{23109632} a^{13} + \frac{1161367}{5777408} a^{11} + \frac{14803}{111104} a^{9} - \frac{150491}{361088} a^{7} - \frac{21325}{45136} a^{5} + \frac{95}{217} a^{3} + \frac{622}{2821} a$, $\frac{1}{4483268608} a^{30} + \frac{23}{560408576} a^{28} - \frac{3}{2694272} a^{26} + \frac{45}{9038848} a^{24} - \frac{3039873}{4483268608} a^{22} + \frac{58955}{21554176} a^{20} - \frac{990237}{70051072} a^{18} + \frac{147580773}{560408576} a^{16} - \frac{171259579}{344866816} a^{14} - \frac{12516013}{140102144} a^{12} - \frac{19337341}{140102144} a^{10} + \frac{866457}{2694272} a^{8} + \frac{455489}{1094548} a^{6} - \frac{1392331}{4378192} a^{4} - \frac{53}{42098} a^{2} + \frac{100882}{273637}$, $\frac{1}{8966537216} a^{31} + \frac{23}{1120817152} a^{29} - \frac{3}{5388544} a^{27} + \frac{45}{18077696} a^{25} - \frac{3039873}{8966537216} a^{23} + \frac{58955}{43108352} a^{21} - \frac{990237}{140102144} a^{19} + \frac{147580773}{1120817152} a^{17} + \frac{173607237}{689733632} a^{15} - \frac{12516013}{280204288} a^{13} - \frac{19337341}{280204288} a^{11} + \frac{866457}{5388544} a^{9} + \frac{455489}{2189096} a^{7} - \frac{1392331}{8756384} a^{5} - \frac{53}{84196} a^{3} + \frac{50441}{273637} a$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( \frac{14177}{140102144} a^{31} - \frac{841741}{2241634304} a^{29} + \frac{31183}{43108352} a^{27} - \frac{20575}{35025536} a^{25} - \frac{30469}{35025536} a^{23} + \frac{897217}{172433408} a^{21} - \frac{6169573}{560408576} a^{19} + \frac{1059733}{70051072} a^{17} - \frac{585987}{21554176} a^{15} - \frac{11507309}{2241634304} a^{13} + \frac{79556065}{560408576} a^{11} - \frac{577719}{1347136} a^{9} + \frac{22725057}{35025536} a^{7} - \frac{761483}{1094548} a^{5} - \frac{11059}{24056} a^{3} + \frac{425238}{273637} a \) (order $48$)
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:  \( 37735500852.51004 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 37735500852.51004 \cdot 12}{48\sqrt{23790908696561643372461609312578223409406672896}}\approx 0.360879743901538$ (assuming GRH)

Galois group

$C_2^2\times C_2^2:C_4$ (as 32T262):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 64
The 40 conjugacy class representatives for $C_2^2\times C_2^2:C_4$
Character table for $C_2^2\times C_2^2:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), 4.4.9792.1, \(\Q(\zeta_{16})^+\), 4.4.18432.1, 4.4.4352.1, 4.0.18432.2, 4.0.1088.2, 4.0.39168.3, 4.0.2048.2, \(\Q(i, \sqrt{6})\), 4.4.313344.1, 4.4.34816.1, 4.0.34816.1, 4.0.313344.1, \(\Q(\zeta_{12})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), 8.8.392737849344.2, 8.0.98184462336.5, 8.0.4848615424.13, 8.0.392737849344.12, 8.0.98184462336.10, 8.0.392737849344.35, \(\Q(\zeta_{24})\), 8.8.98184462336.2, 8.8.392737849344.1, 8.0.98184462336.11, 8.0.392737849344.42, 8.8.98184462336.1, 8.8.4848615424.1, 8.0.392737849344.37, 8.0.1212153856.10, 8.0.98184462336.37, 8.0.1212153856.9, 8.0.392737849344.40, 8.0.4848615424.10, 8.0.392737849344.24, 8.0.98184462336.12, 8.0.392737849344.45, 8.0.98184462336.46, 8.0.1358954496.4, 8.0.1534132224.8, 8.0.18939904.2, \(\Q(\zeta_{16})\), 8.0.339738624.2, 8.0.95883264.1, 8.0.1534132224.10, 8.0.339738624.1, \(\Q(\zeta_{48})^+\), 8.8.1534132224.1, 8.0.1534132224.4, 8.0.1358954496.3, Deg 16, 16.16.154243018307350441230336.1, Deg 16, 16.0.9640188644209402576896.1, Deg 16, Deg 16, 16.0.23509071529850699776.10, Deg 16, Deg 16, Deg 16, 16.0.9640188644209402576896.2, 16.0.154243018307350441230336.1, Deg 16, \(\Q(\zeta_{48})\), 16.0.2353561680715186176.2

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$