Properties

Label 32.0.146...776.1
Degree $32$
Signature $[0, 16]$
Discriminant $1.465\times 10^{46}$
Root discriminant $27.71$
Ramified primes $2, 3$
Class number $9$ (GRH)
Class group $[3, 3]$ (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 - x^16 + 1)
 
gp: K = bnfinit(x^32 - 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, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 

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

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:  \(14648040110065267094876580444599852735215435776\)\(\medspace = 2^{128}\cdot 3^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $27.71$
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:  $2, 3$
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:  \(96=2^{5}\cdot 3\)
Dirichlet character group:    $\lbrace$$\chi_{96}(1,·)$, $\chi_{96}(5,·)$, $\chi_{96}(7,·)$, $\chi_{96}(11,·)$, $\chi_{96}(13,·)$, $\chi_{96}(17,·)$, $\chi_{96}(19,·)$, $\chi_{96}(23,·)$, $\chi_{96}(25,·)$, $\chi_{96}(29,·)$, $\chi_{96}(31,·)$, $\chi_{96}(35,·)$, $\chi_{96}(37,·)$, $\chi_{96}(41,·)$, $\chi_{96}(43,·)$, $\chi_{96}(47,·)$, $\chi_{96}(49,·)$, $\chi_{96}(53,·)$, $\chi_{96}(55,·)$, $\chi_{96}(59,·)$, $\chi_{96}(61,·)$, $\chi_{96}(65,·)$, $\chi_{96}(67,·)$, $\chi_{96}(71,·)$, $\chi_{96}(73,·)$, $\chi_{96}(77,·)$, $\chi_{96}(79,·)$, $\chi_{96}(83,·)$, $\chi_{96}(85,·)$, $\chi_{96}(89,·)$, $\chi_{96}(91,·)$, $\chi_{96}(95,·)$$\rbrace$
This is a CM field.

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

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

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( a \) (order $96$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( a^{6} + a^{3} + 1 \),  \( a^{18} + a^{9} + 1 \),  \( a^{12} + a^{9} + a^{6} + a^{3} + 1 \),  \( a^{26} - a^{10} - a^{6} + 1 \),  \( a^{26} - a^{10} - a^{6} - 1 \),  \( a^{20} - 1 \),  \( a^{18} + a^{14} - a^{2} \),  \( a^{10} - 1 \),  \( a^{5} - 1 \),  \( a^{13} - 1 \),  \( a^{30} - a^{18} - 1 \),  \( a^{18} - a^{2} - 1 \),  \( a^{18} - a^{17} + a - 1 \),  \( a - 1 \),  \( a^{7} + 1 \) (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 47233168152.25636 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 47233168152.25636 \cdot 9}{96\sqrt{14648040110065267094876580444599852735215435776}}\approx 0.215877055915058$ (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\zeta_{12})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\zeta_{16})^+\), 4.0.18432.2, 4.0.2048.2, 4.4.18432.1, \(\Q(\zeta_{24})\), 8.0.339738624.2, 8.0.339738624.1, \(\Q(\zeta_{16})\), 8.0.1358954496.4, \(\Q(\zeta_{48})^+\), 8.0.1358954496.3, \(\Q(\zeta_{32})^+\), 8.0.173946175488.1, 8.0.2147483648.1, 8.8.173946175488.1, \(\Q(\zeta_{48})\), 16.0.30257271966902092038144.1, 16.0.30257271966902092038144.2, \(\Q(\zeta_{32})\), 16.0.121029087867608368152576.2, \(\Q(\zeta_{96})^+\), 16.0.121029087867608368152576.1

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
3Data not computed