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Properties

Label 16.0.11268182541...5233.2
Degree $16$
Signature $[0, 8]$
Discriminant $17^{15}\cdot 89^{8}$
Root discriminant $134.35$
Ramified primes $17, 89$
Class number $2965264$ (GRH)
Class group $[2, 2, 2, 370658]$ (GRH)
Galois group $C_{16}$ (as 16T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1528698279571, -595808429459, 595808429459, -86959420307, 86959420307, -6006168851, 6006168851, -223793747, 223793747, -4764387, 4764387, -57971, 57971, -375, 375, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 375*x^14 - 375*x^13 + 57971*x^12 - 57971*x^11 + 4764387*x^10 - 4764387*x^9 + 223793747*x^8 - 223793747*x^7 + 6006168851*x^6 - 6006168851*x^5 + 86959420307*x^4 - 86959420307*x^3 + 595808429459*x^2 - 595808429459*x + 1528698279571)
 
gp: K = bnfinit(x^16 - x^15 + 375*x^14 - 375*x^13 + 57971*x^12 - 57971*x^11 + 4764387*x^10 - 4764387*x^9 + 223793747*x^8 - 223793747*x^7 + 6006168851*x^6 - 6006168851*x^5 + 86959420307*x^4 - 86959420307*x^3 + 595808429459*x^2 - 595808429459*x + 1528698279571, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 375 x^{14} - 375 x^{13} + 57971 x^{12} - 57971 x^{11} + 4764387 x^{10} - 4764387 x^{9} + 223793747 x^{8} - 223793747 x^{7} + 6006168851 x^{6} - 6006168851 x^{5} + 86959420307 x^{4} - 86959420307 x^{3} + 595808429459 x^{2} - 595808429459 x + 1528698279571 \)

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:  \(11268182541757132036763060346765233=17^{15}\cdot 89^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $134.35$
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:  $17, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1513=17\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{1513}(1,·)$, $\chi_{1513}(1158,·)$, $\chi_{1513}(711,·)$, $\chi_{1513}(713,·)$, $\chi_{1513}(266,·)$, $\chi_{1513}(268,·)$, $\chi_{1513}(1423,·)$, $\chi_{1513}(533,·)$, $\chi_{1513}(535,·)$, $\chi_{1513}(88,·)$, $\chi_{1513}(1069,·)$, $\chi_{1513}(622,·)$, $\chi_{1513}(177,·)$, $\chi_{1513}(179,·)$, $\chi_{1513}(889,·)$, $\chi_{1513}(446,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{170982561751} a^{9} + \frac{14616106291}{170982561751} a^{8} + \frac{198}{170982561751} a^{7} + \frac{7696280951}{170982561751} a^{6} + \frac{13068}{170982561751} a^{5} + \frac{81330328803}{170982561751} a^{4} + \frac{319440}{170982561751} a^{3} - \frac{73232756476}{170982561751} a^{2} + \frac{2108304}{170982561751} a - \frac{30407518558}{170982561751}$, $\frac{1}{170982561751} a^{10} + \frac{220}{170982561751} a^{8} + \frac{20410785100}{170982561751} a^{7} + \frac{16940}{170982561751} a^{6} + \frac{65574793882}{170982561751} a^{5} + \frac{532400}{170982561751} a^{4} - \frac{21412618959}{170982561751} a^{3} + \frac{5856400}{170982561751} a^{2} - \frac{64556246798}{170982561751} a + \frac{10307264}{170982561751}$, $\frac{1}{170982561751} a^{11} + \frac{53536074349}{170982561751} a^{8} - \frac{26620}{170982561751} a^{7} + \frac{82218602172}{170982561751} a^{6} - \frac{2342560}{170982561751} a^{5} + \frac{39084028236}{170982561751} a^{4} - \frac{64420400}{170982561751} a^{3} - \frac{25710626672}{170982561751} a^{2} - \frac{453519616}{170982561751} a + \frac{21334174471}{170982561751}$, $\frac{1}{170982561751} a^{12} - \frac{31944}{170982561751} a^{8} + \frac{82994709632}{170982561751} a^{7} - \frac{3279584}{170982561751} a^{6} - \frac{80675441155}{170982561751} a^{5} - \frac{115956720}{170982561751} a^{4} - \frac{84456897963}{170982561751} a^{3} - \frac{1360558848}{170982561751} a^{2} + \frac{78163444503}{170982561751} a - \frac{2494357888}{170982561751}$, $\frac{1}{170982561751} a^{13} + \frac{26517927355}{170982561751} a^{8} + \frac{3045328}{170982561751} a^{7} + \frac{67382021402}{170982561751} a^{6} + \frac{301487472}{170982561751} a^{5} + \frac{22523140375}{170982561751} a^{4} + \frac{8843632512}{170982561751} a^{3} - \frac{56582109410}{170982561751} a^{2} + \frac{64853305088}{170982561751} a + \frac{14160490679}{170982561751}$, $\frac{1}{170982561751} a^{14} + \frac{3875872}{170982561751} a^{8} - \frac{53690742358}{170982561751} a^{7} + \frac{447663216}{170982561751} a^{6} + \frac{67901134512}{170982561751} a^{5} + \frac{16883298432}{170982561751} a^{4} + \frac{45760439183}{170982561751} a^{3} + \frac{35368863529}{170982561751} a^{2} + \frac{39887576739}{170982561751} a + \frac{47154707026}{170982561751}$, $\frac{1}{170982561751} a^{15} + \frac{73511411712}{170982561751} a^{8} - \frac{319759440}{170982561751} a^{7} + \frac{562026351}{1692896651} a^{6} - \frac{33766596864}{170982561751} a^{5} - \frac{53797893919}{170982561751} a^{4} - \frac{5861755894}{170982561751} a^{3} + \frac{31683044004}{170982561751} a^{2} + \frac{82801229986}{170982561751} a - \frac{65308105459}{170982561751}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{370658}$, which has order $2965264$ (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:  \( 3640.01221338 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{16}$ (as 16T1):

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

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
$89$89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$