Properties

Label 32.0.352...129.1
Degree $32$
Signature $[0, 16]$
Discriminant $3.527\times 10^{44}$
Root discriminant $24.67$
Ramified primes $3, 17$
Class number $5$ (GRH)
Class group $[5]$ (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^29 - x^28 + x^26 - x^25 + x^23 - x^22 + x^20 - x^19 + x^17 - x^16 + x^15 - x^13 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1)
 
gp: K = bnfinit(x^32 - x^31 + x^29 - x^28 + x^26 - x^25 + x^23 - x^22 + x^20 - x^19 + x^17 - x^16 + x^15 - x^13 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1]);
 

\( x^{32} - x^{31} + x^{29} - x^{28} + x^{26} - x^{25} + x^{23} - x^{22} + x^{20} - x^{19} + x^{17} - x^{16} + x^{15} - x^{13} + x^{12} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \)

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:  \(352701833122210710593389803720131611763844129\)\(\medspace = 3^{16}\cdot 17^{30}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $24.67$
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:  $3, 17$
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:  \(51=3\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{51}(1,·)$, $\chi_{51}(2,·)$, $\chi_{51}(4,·)$, $\chi_{51}(5,·)$, $\chi_{51}(7,·)$, $\chi_{51}(8,·)$, $\chi_{51}(10,·)$, $\chi_{51}(11,·)$, $\chi_{51}(13,·)$, $\chi_{51}(14,·)$, $\chi_{51}(16,·)$, $\chi_{51}(19,·)$, $\chi_{51}(20,·)$, $\chi_{51}(22,·)$, $\chi_{51}(23,·)$, $\chi_{51}(25,·)$, $\chi_{51}(26,·)$, $\chi_{51}(28,·)$, $\chi_{51}(29,·)$, $\chi_{51}(31,·)$, $\chi_{51}(32,·)$, $\chi_{51}(35,·)$, $\chi_{51}(37,·)$, $\chi_{51}(38,·)$, $\chi_{51}(40,·)$, $\chi_{51}(41,·)$, $\chi_{51}(43,·)$, $\chi_{51}(44,·)$, $\chi_{51}(46,·)$, $\chi_{51}(47,·)$, $\chi_{51}(49,·)$, $\chi_{51}(50,·)$$\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

$C_{5}$, which has order $5$ (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 $102$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( a^{3} + 1 \),  \( a^{6} + 1 \),  \( a^{21} - a^{19} + a^{6} - a^{2} \),  \( a^{19} - a^{18} + a^{2} - 1 \),  \( a^{6} + a^{3} + 1 \),  \( a^{18} + a^{15} + a^{12} + a^{9} + a^{6} + a^{3} + 1 \),  \( a^{27} + a^{3} \),  \( a - 1 \),  \( a^{2} - 1 \),  \( a^{4} - 1 \),  \( a^{8} - 1 \),  \( a^{5} - 1 \),  \( a^{7} - 1 \),  \( a^{11} - 1 \),  \( a^{14} - 1 \) (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 9977645145.96466 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 9977645145.96466 \cdot 5}{102\sqrt{352701833122210710593389803720131611763844129}}\approx 0.153663925012584$ (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{-3}, \sqrt{17})\), 4.4.4913.1, 4.0.44217.1, 8.0.1955143089.1, \(\Q(\zeta_{17})^+\), 8.0.33237432513.1, 16.0.1104726920056229495169.1, \(\Q(\zeta_{17})\), \(\Q(\zeta_{51})^+\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{4}$ R $16^{2}$ $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ 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
3Data not computed
17Data not computed