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Properties

Label 16.0.28621836185...0801.7
Degree $16$
Signature $[0, 8]$
Discriminant $17^{12}\cdot 53^{12}$
Root discriminant $164.45$
Ramified primes $17, 53$
Class number $13328$ (GRH)
Class group $[2, 14, 476]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![68049763, -41607389, -42465958, 40577113, 25631015, -1201029, -510377, 363819, -100624, 5215, 4773, -3772, 450, 14, 1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + x^14 + 14*x^13 + 450*x^12 - 3772*x^11 + 4773*x^10 + 5215*x^9 - 100624*x^8 + 363819*x^7 - 510377*x^6 - 1201029*x^5 + 25631015*x^4 + 40577113*x^3 - 42465958*x^2 - 41607389*x + 68049763)
 
gp: K = bnfinit(x^16 - 4*x^15 + x^14 + 14*x^13 + 450*x^12 - 3772*x^11 + 4773*x^10 + 5215*x^9 - 100624*x^8 + 363819*x^7 - 510377*x^6 - 1201029*x^5 + 25631015*x^4 + 40577113*x^3 - 42465958*x^2 - 41607389*x + 68049763, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + x^{14} + 14 x^{13} + 450 x^{12} - 3772 x^{11} + 4773 x^{10} + 5215 x^{9} - 100624 x^{8} + 363819 x^{7} - 510377 x^{6} - 1201029 x^{5} + 25631015 x^{4} + 40577113 x^{3} - 42465958 x^{2} - 41607389 x + 68049763 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(286218361857094751614277009933470801=17^{12}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $164.45$
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:  $17, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{33618820} a^{12} - \frac{3}{33618820} a^{11} + \frac{400819}{33618820} a^{10} - \frac{5330869}{33618820} a^{9} - \frac{3057511}{16809410} a^{8} + \frac{12325573}{33618820} a^{7} - \frac{2853223}{16809410} a^{6} - \frac{1739309}{33618820} a^{5} + \frac{114848}{8404705} a^{4} + \frac{6978509}{33618820} a^{3} - \frac{3546531}{33618820} a^{2} - \frac{3560687}{8404705} a + \frac{12406957}{33618820}$, $\frac{1}{33618820} a^{13} + \frac{40081}{3361882} a^{11} - \frac{1032103}{8404705} a^{10} - \frac{5298219}{33618820} a^{9} - \frac{6019493}{33618820} a^{8} - \frac{2348547}{33618820} a^{7} + \frac{14760173}{33618820} a^{6} + \frac{2410175}{6723764} a^{5} + \frac{1671337}{6723764} a^{4} + \frac{289793}{16809410} a^{3} - \frac{8072931}{33618820} a^{2} - \frac{13511877}{33618820} a + \frac{3602051}{33618820}$, $\frac{1}{33618820} a^{14} - \frac{1462991}{16809410} a^{11} + \frac{6779171}{33618820} a^{10} - \frac{1330113}{33618820} a^{9} - \frac{3643417}{33618820} a^{8} - \frac{13411417}{33618820} a^{7} - \frac{2025549}{6723764} a^{6} + \frac{827209}{6723764} a^{5} - \frac{3715051}{8404705} a^{4} + \frac{4749369}{33618820} a^{3} + \frac{630993}{33618820} a^{2} - \frac{4302169}{33618820} a + \frac{649550}{1680941}$, $\frac{1}{128903995978246149644883815153301740} a^{15} - \frac{957152561001054635773816417}{64451997989123074822441907576650870} a^{14} + \frac{17090874604512178594850763}{1371319106151554783456210799503210} a^{13} - \frac{207398812596108694435568539}{25780799195649229928976763030660348} a^{12} - \frac{13313871886094433156962875506494151}{64451997989123074822441907576650870} a^{11} + \frac{13311572857651370598804119314016981}{64451997989123074822441907576650870} a^{10} - \frac{7140804190937573060669629413184104}{32225998994561537411220953788325435} a^{9} + \frac{13492056664613857755055135677242421}{128903995978246149644883815153301740} a^{8} - \frac{953350184304250674850397759148601}{12890399597824614964488381515330174} a^{7} + \frac{39877544190025210873624960816358819}{128903995978246149644883815153301740} a^{6} - \frac{28743255219023341081538624580739607}{128903995978246149644883815153301740} a^{5} + \frac{3840153530334158282570291132572919}{128903995978246149644883815153301740} a^{4} + \frac{26418989622772188607365198336501601}{64451997989123074822441907576650870} a^{3} - \frac{283423585384837519695283031112637}{12890399597824614964488381515330174} a^{2} - \frac{13957187086629797366375871494090946}{32225998994561537411220953788325435} a - \frac{10902910722969133484178069925618879}{128903995978246149644883815153301740}$

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

Class group and class number

$C_{2}\times C_{14}\times C_{476}$, which has order $13328$ (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:  $7$
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:  \( 6878292.5579 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{53}) \), \(\Q(\sqrt{901}) \), \(\Q(\sqrt{17}) \), 4.0.148877.1, \(\Q(\sqrt{17}, \sqrt{53})\), 4.0.43025453.2, 4.4.13800617.1 x2, 4.4.260389.1 x2, 8.0.1851189605855209.2, 8.8.190457029580689.1, 8.0.534993796092155401.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$53$53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$