Properties

Label 32.0.125...625.1
Degree $32$
Signature $[0, 16]$
Discriminant $1.250\times 10^{48}$
Root discriminant $31.84$
Ramified primes $5, 17$
Class number $17$ (GRH)
Class group $[17]$ (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 + 2*x^30 - 3*x^29 + 5*x^28 - 8*x^27 + 13*x^26 - 21*x^25 + 34*x^24 - 55*x^23 + 89*x^22 - 144*x^21 + 233*x^20 - 377*x^19 + 610*x^18 - 987*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1)
 
gp: K = bnfinit(x^32 - x^31 + 2*x^30 - 3*x^29 + 5*x^28 - 8*x^27 + 13*x^26 - 21*x^25 + 34*x^24 - 55*x^23 + 89*x^22 - 144*x^21 + 233*x^20 - 377*x^19 + 610*x^18 - 987*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, -987, 610, -377, 233, -144, 89, -55, 34, -21, 13, -8, 5, -3, 2, -1, 1]);
 

\( x^{32} - x^{31} + 2 x^{30} - 3 x^{29} + 5 x^{28} - 8 x^{27} + 13 x^{26} - 21 x^{25} + 34 x^{24} - 55 x^{23} + 89 x^{22} - 144 x^{21} + 233 x^{20} - 377 x^{19} + 610 x^{18} - 987 x^{17} + 1597 x^{16} + 987 x^{15} + 610 x^{14} + 377 x^{13} + 233 x^{12} + 144 x^{11} + 89 x^{10} + 55 x^{9} + 34 x^{8} + 21 x^{7} + 13 x^{6} + 8 x^{5} + 5 x^{4} + 3 x^{3} + 2 x^{2} + x + 1 \)

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:  \(1250223652010309685753479160887484565582275390625\)\(\medspace = 5^{16}\cdot 17^{30}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $31.84$
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:  $5, 17$
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:  \(85=5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{85}(1,·)$, $\chi_{85}(4,·)$, $\chi_{85}(6,·)$, $\chi_{85}(9,·)$, $\chi_{85}(11,·)$, $\chi_{85}(14,·)$, $\chi_{85}(16,·)$, $\chi_{85}(19,·)$, $\chi_{85}(21,·)$, $\chi_{85}(24,·)$, $\chi_{85}(26,·)$, $\chi_{85}(29,·)$, $\chi_{85}(31,·)$, $\chi_{85}(36,·)$, $\chi_{85}(39,·)$, $\chi_{85}(41,·)$, $\chi_{85}(44,·)$, $\chi_{85}(46,·)$, $\chi_{85}(49,·)$, $\chi_{85}(54,·)$, $\chi_{85}(56,·)$, $\chi_{85}(59,·)$, $\chi_{85}(61,·)$, $\chi_{85}(64,·)$, $\chi_{85}(66,·)$, $\chi_{85}(69,·)$, $\chi_{85}(71,·)$, $\chi_{85}(74,·)$, $\chi_{85}(76,·)$, $\chi_{85}(79,·)$, $\chi_{85}(81,·)$, $\chi_{85}(84,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1597} a^{17} - \frac{610}{1597}$, $\frac{1}{1597} a^{18} - \frac{610}{1597} a$, $\frac{1}{1597} a^{19} - \frac{610}{1597} a^{2}$, $\frac{1}{1597} a^{20} - \frac{610}{1597} a^{3}$, $\frac{1}{1597} a^{21} - \frac{610}{1597} a^{4}$, $\frac{1}{1597} a^{22} - \frac{610}{1597} a^{5}$, $\frac{1}{1597} a^{23} - \frac{610}{1597} a^{6}$, $\frac{1}{1597} a^{24} - \frac{610}{1597} a^{7}$, $\frac{1}{1597} a^{25} - \frac{610}{1597} a^{8}$, $\frac{1}{1597} a^{26} - \frac{610}{1597} a^{9}$, $\frac{1}{1597} a^{27} - \frac{610}{1597} a^{10}$, $\frac{1}{1597} a^{28} - \frac{610}{1597} a^{11}$, $\frac{1}{1597} a^{29} - \frac{610}{1597} a^{12}$, $\frac{1}{1597} a^{30} - \frac{610}{1597} a^{13}$, $\frac{1}{1597} a^{31} - \frac{610}{1597} a^{14}$

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:  \( -\frac{13}{1597} a^{24} - \frac{46368}{1597} a^{7} \) (order $34$)
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:  \( 42597284566.379654 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 42597284566.379654 \cdot 17}{34\sqrt{1250223652010309685753479160887484565582275390625}}\approx 0.112392144505091$ (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

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{17})\), 4.4.4913.1, 4.4.122825.1, 8.8.15085980625.1, \(\Q(\zeta_{17})^+\), 8.8.256461670625.1, 16.16.65772588499765987890625.1, \(\Q(\zeta_{17})\), 16.0.1118134004496021794140625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{4}$ $16^{2}$ R $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ 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
5Data not computed
17Data not computed