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Properties

Label 32.32.5369669698...0000.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{32}\cdot 5^{16}\cdot 17^{30}$
Root discriminant $63.69$
Ramified primes $2, 5, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_{16}$ (as 32T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![289, 0, -10404, 0, 126582, 0, -780300, 0, 2859077, 0, -6760866, 0, 10821316, 0, -12057369, 0, 9493310, 0, -5308080, 0, 2100605, 0, -581910, 0, 110704, 0, -14025, 0, 1122, 0, -51, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 51*x^30 + 1122*x^28 - 14025*x^26 + 110704*x^24 - 581910*x^22 + 2100605*x^20 - 5308080*x^18 + 9493310*x^16 - 12057369*x^14 + 10821316*x^12 - 6760866*x^10 + 2859077*x^8 - 780300*x^6 + 126582*x^4 - 10404*x^2 + 289)
 
gp: K = bnfinit(x^32 - 51*x^30 + 1122*x^28 - 14025*x^26 + 110704*x^24 - 581910*x^22 + 2100605*x^20 - 5308080*x^18 + 9493310*x^16 - 12057369*x^14 + 10821316*x^12 - 6760866*x^10 + 2859077*x^8 - 780300*x^6 + 126582*x^4 - 10404*x^2 + 289, 1)
 

Normalized defining polynomial

\( x^{32} - 51 x^{30} + 1122 x^{28} - 14025 x^{26} + 110704 x^{24} - 581910 x^{22} + 2100605 x^{20} - 5308080 x^{18} + 9493310 x^{16} - 12057369 x^{14} + 10821316 x^{12} - 6760866 x^{10} + 2859077 x^{8} - 780300 x^{6} + 126582 x^{4} - 10404 x^{2} + 289 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[32, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5369669698069964755143230114229268544880640000000000000000=2^{32}\cdot 5^{16}\cdot 17^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.69$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(340=2^{2}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{340}(1,·)$, $\chi_{340}(131,·)$, $\chi_{340}(9,·)$, $\chi_{340}(11,·)$, $\chi_{340}(21,·)$, $\chi_{340}(279,·)$, $\chi_{340}(281,·)$, $\chi_{340}(31,·)$, $\chi_{340}(161,·)$, $\chi_{340}(39,·)$, $\chi_{340}(169,·)$, $\chi_{340}(299,·)$, $\chi_{340}(49,·)$, $\chi_{340}(311,·)$, $\chi_{340}(159,·)$, $\chi_{340}(189,·)$, $\chi_{340}(321,·)$, $\chi_{340}(139,·)$, $\chi_{340}(69,·)$, $\chi_{340}(71,·)$, $\chi_{340}(79,·)$, $\chi_{340}(81,·)$, $\chi_{340}(211,·)$, $\chi_{340}(89,·)$, $\chi_{340}(91,·)$, $\chi_{340}(199,·)$, $\chi_{340}(101,·)$, $\chi_{340}(99,·)$, $\chi_{340}(229,·)$, $\chi_{340}(231,·)$, $\chi_{340}(121,·)$, $\chi_{340}(149,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{17} a^{16}$, $\frac{1}{17} a^{17}$, $\frac{1}{17} a^{18}$, $\frac{1}{17} a^{19}$, $\frac{1}{17} a^{20}$, $\frac{1}{17} a^{21}$, $\frac{1}{17} a^{22}$, $\frac{1}{17} a^{23}$, $\frac{1}{68} a^{24} - \frac{1}{68} a^{18} + \frac{1}{4} a^{12} - \frac{1}{4} a^{6} + \frac{1}{4}$, $\frac{1}{68} a^{25} - \frac{1}{68} a^{19} + \frac{1}{4} a^{13} - \frac{1}{4} a^{7} + \frac{1}{4} a$, $\frac{1}{68} a^{26} - \frac{1}{68} a^{20} + \frac{1}{4} a^{14} - \frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{68} a^{27} - \frac{1}{68} a^{21} + \frac{1}{4} a^{15} - \frac{1}{4} a^{9} + \frac{1}{4} a^{3}$, $\frac{1}{6868} a^{28} + \frac{3}{6868} a^{26} - \frac{7}{1717} a^{24} - \frac{81}{6868} a^{22} - \frac{3}{404} a^{20} - \frac{14}{1717} a^{18} - \frac{139}{6868} a^{16} - \frac{173}{404} a^{14} - \frac{44}{101} a^{12} - \frac{117}{404} a^{10} + \frac{117}{404} a^{8} + \frac{20}{101} a^{6} + \frac{9}{404} a^{4} + \frac{15}{404} a^{2} + \frac{21}{101}$, $\frac{1}{6868} a^{29} + \frac{3}{6868} a^{27} - \frac{7}{1717} a^{25} - \frac{81}{6868} a^{23} - \frac{3}{404} a^{21} - \frac{14}{1717} a^{19} - \frac{139}{6868} a^{17} - \frac{173}{404} a^{15} - \frac{44}{101} a^{13} - \frac{117}{404} a^{11} + \frac{117}{404} a^{9} + \frac{20}{101} a^{7} + \frac{9}{404} a^{5} + \frac{15}{404} a^{3} + \frac{21}{101} a$, $\frac{1}{97501863786788} a^{30} + \frac{4070710253}{97501863786788} a^{28} - \frac{18227922359}{5735403752164} a^{26} - \frac{532467546049}{97501863786788} a^{24} - \frac{1527763521993}{97501863786788} a^{22} - \frac{1851473411465}{97501863786788} a^{20} - \frac{573860201243}{97501863786788} a^{18} + \frac{2208982539189}{97501863786788} a^{16} + \frac{511045344477}{5735403752164} a^{14} - \frac{1441046843757}{5735403752164} a^{12} + \frac{2738967897743}{5735403752164} a^{10} - \frac{2573739796453}{5735403752164} a^{8} + \frac{2240246303933}{5735403752164} a^{6} - \frac{2364746259999}{5735403752164} a^{4} - \frac{1945968873747}{5735403752164} a^{2} + \frac{607380117581}{1433850938041}$, $\frac{1}{97501863786788} a^{31} + \frac{4070710253}{97501863786788} a^{29} - \frac{18227922359}{5735403752164} a^{27} - \frac{532467546049}{97501863786788} a^{25} - \frac{1527763521993}{97501863786788} a^{23} - \frac{1851473411465}{97501863786788} a^{21} - \frac{573860201243}{97501863786788} a^{19} + \frac{2208982539189}{97501863786788} a^{17} + \frac{511045344477}{5735403752164} a^{15} - \frac{1441046843757}{5735403752164} a^{13} + \frac{2738967897743}{5735403752164} a^{11} - \frac{2573739796453}{5735403752164} a^{9} + \frac{2240246303933}{5735403752164} a^{7} - \frac{2364746259999}{5735403752164} a^{5} - \frac{1945968873747}{5735403752164} a^{3} + \frac{607380117581}{1433850938041} a$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $31$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4821130960706681000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{16}$ (as 32T32):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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, \(\Q(\zeta_{68})^+\), 16.16.73278030118651284300800000000.1

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 $16^{2}$ 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.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
5Data not computed
17Data not computed