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Properties

Label 32.0.519...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $5.192\times 10^{49}$
Root discriminant $35.78$
Ramified primes $2, 5$
Class number $17$ (GRH)
Class group $[17]$ (GRH)
Galois group $C_2^2\times C_8$ (as 32T37)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 2207*x^16 + 1)
 
gp: K = bnfinit(x^32 + 2207*x^16 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2207, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 

\( x^{32} + 2207 x^{16} + 1 \)

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:  \(51922968585348276285304963292200960000000000000000\)\(\medspace = 2^{128}\cdot 5^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $35.78$
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:  $2, 5$
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:  \(160=2^{5}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{160}(1,·)$, $\chi_{160}(131,·)$, $\chi_{160}(129,·)$, $\chi_{160}(9,·)$, $\chi_{160}(11,·)$, $\chi_{160}(141,·)$, $\chi_{160}(19,·)$, $\chi_{160}(21,·)$, $\chi_{160}(151,·)$, $\chi_{160}(29,·)$, $\chi_{160}(159,·)$, $\chi_{160}(39,·)$, $\chi_{160}(41,·)$, $\chi_{160}(49,·)$, $\chi_{160}(51,·)$, $\chi_{160}(31,·)$, $\chi_{160}(61,·)$, $\chi_{160}(139,·)$, $\chi_{160}(69,·)$, $\chi_{160}(71,·)$, $\chi_{160}(79,·)$, $\chi_{160}(81,·)$, $\chi_{160}(89,·)$, $\chi_{160}(91,·)$, $\chi_{160}(99,·)$, $\chi_{160}(101,·)$, $\chi_{160}(109,·)$, $\chi_{160}(111,·)$, $\chi_{160}(59,·)$, $\chi_{160}(119,·)$, $\chi_{160}(121,·)$, $\chi_{160}(149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{987} a^{16} - \frac{377}{987}$, $\frac{1}{987} a^{17} - \frac{377}{987} a$, $\frac{1}{987} a^{18} - \frac{377}{987} a^{2}$, $\frac{1}{987} a^{19} - \frac{377}{987} a^{3}$, $\frac{1}{987} a^{20} - \frac{377}{987} a^{4}$, $\frac{1}{987} a^{21} - \frac{377}{987} a^{5}$, $\frac{1}{987} a^{22} - \frac{377}{987} a^{6}$, $\frac{1}{987} a^{23} - \frac{377}{987} a^{7}$, $\frac{1}{987} a^{24} - \frac{377}{987} a^{8}$, $\frac{1}{987} a^{25} - \frac{377}{987} a^{9}$, $\frac{1}{987} a^{26} - \frac{377}{987} a^{10}$, $\frac{1}{987} a^{27} - \frac{377}{987} a^{11}$, $\frac{1}{987} a^{28} - \frac{377}{987} a^{12}$, $\frac{1}{987} a^{29} - \frac{377}{987} a^{13}$, $\frac{1}{987} a^{30} - \frac{377}{987} a^{14}$, $\frac{1}{987} a^{31} - \frac{377}{987} a^{15}$

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

Class group and class number

$C_{17}$, which has order $17$ (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:  \( \frac{5}{987} a^{21} + \frac{10946}{987} a^{5} \) (order $32$)
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:  \( 323273529801.8212 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 323273529801.8212 \cdot 17}{32\sqrt{51922968585348276285304963292200960000000000000000}}\approx 0.140626518450650$ (assuming GRH)

Galois group

$C_2^2\times C_8$ (as 32T37):

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 $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{10}) \), \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\zeta_{16})^+\), 4.0.2048.2, 4.0.51200.2, 4.4.51200.1, 8.0.40960000.1, \(\Q(\zeta_{16})\), 8.0.10485760000.3, 8.0.10485760000.2, 8.0.10485760000.1, 8.8.2621440000.1, 8.0.2621440000.1, 8.8.1342177280000.1, 8.0.1342177280000.1, 8.0.2147483648.1, \(\Q(\zeta_{32})^+\), 16.0.109951162777600000000.1, 16.0.7205759403792793600000000.2, \(\Q(\zeta_{32})\), 16.0.7205759403792793600000000.3, 16.0.7205759403792793600000000.1, 16.16.1801439850948198400000000.1, 16.0.1801439850948198400000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
2Data not computed
5Data not computed