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Properties

Label 16.0.62542703776...4257.1
Degree $16$
Signature $[0, 8]$
Discriminant $113^{15}$
Root discriminant $84.09$
Ramified prime $113$
Class number $17$ (GRH)
Class group $[17]$ (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![132937, 281708, 210865, 132177, 154492, 119403, 64244, 21729, 12062, 425, 325, -525, 110, -20, 4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 4*x^14 - 20*x^13 + 110*x^12 - 525*x^11 + 325*x^10 + 425*x^9 + 12062*x^8 + 21729*x^7 + 64244*x^6 + 119403*x^5 + 154492*x^4 + 132177*x^3 + 210865*x^2 + 281708*x + 132937)
 
gp: K = bnfinit(x^16 - x^15 + 4*x^14 - 20*x^13 + 110*x^12 - 525*x^11 + 325*x^10 + 425*x^9 + 12062*x^8 + 21729*x^7 + 64244*x^6 + 119403*x^5 + 154492*x^4 + 132177*x^3 + 210865*x^2 + 281708*x + 132937, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 4 x^{14} - 20 x^{13} + 110 x^{12} - 525 x^{11} + 325 x^{10} + 425 x^{9} + 12062 x^{8} + 21729 x^{7} + 64244 x^{6} + 119403 x^{5} + 154492 x^{4} + 132177 x^{3} + 210865 x^{2} + 281708 x + 132937 \)

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:  \(6254270377697284219642783674257=113^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.09$
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:  $113$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(113\)
Dirichlet character group:    $\lbrace$$\chi_{113}(1,·)$, $\chi_{113}(69,·)$, $\chi_{113}(65,·)$, $\chi_{113}(73,·)$, $\chi_{113}(78,·)$, $\chi_{113}(15,·)$, $\chi_{113}(18,·)$, $\chi_{113}(95,·)$, $\chi_{113}(48,·)$, $\chi_{113}(98,·)$, $\chi_{113}(35,·)$, $\chi_{113}(40,·)$, $\chi_{113}(42,·)$, $\chi_{113}(71,·)$, $\chi_{113}(44,·)$, $\chi_{113}(112,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{436} a^{14} - \frac{63}{436} a^{13} - \frac{19}{436} a^{12} - \frac{53}{436} a^{11} + \frac{3}{436} a^{10} - \frac{5}{218} a^{9} - \frac{40}{109} a^{8} + \frac{149}{436} a^{7} - \frac{41}{218} a^{6} + \frac{63}{218} a^{5} - \frac{14}{109} a^{4} - \frac{53}{436} a^{3} - \frac{105}{218} a^{2} - \frac{81}{218} a - \frac{183}{436}$, $\frac{1}{235507558683255484545564323362469978248} a^{15} - \frac{14071060635376120603235087617011681}{117753779341627742272782161681234989124} a^{14} - \frac{19774378854751556123867676883819401601}{117753779341627742272782161681234989124} a^{13} + \frac{27153008353246984574348125706466443299}{117753779341627742272782161681234989124} a^{12} - \frac{4868728891323672600567396655160185222}{29438444835406935568195540420308747281} a^{11} + \frac{321763238493879405644288355311827049}{33643936954750783506509189051781425464} a^{10} + \frac{11061418121826024317407548199722734389}{117753779341627742272782161681234989124} a^{9} + \frac{26174028567598520913037363987701895451}{235507558683255484545564323362469978248} a^{8} - \frac{116864357488514869615819626621640838237}{235507558683255484545564323362469978248} a^{7} - \frac{12801241542181110715663802090025119751}{117753779341627742272782161681234989124} a^{6} + \frac{5141650721093857519093677813738597581}{117753779341627742272782161681234989124} a^{5} + \frac{640687060262212289925980090238433857}{2160619804433536555463892874885045672} a^{4} + \frac{44327295169702113080387214847158590263}{235507558683255484545564323362469978248} a^{3} - \frac{50427048860278160603440402310424265113}{117753779341627742272782161681234989124} a^{2} - \frac{40992757657569973097801566908220670865}{235507558683255484545564323362469978248} a - \frac{9062047032168468153879991654000985497}{33643936954750783506509189051781425464}$

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

Class group and class number

$C_{17}$, which has order $17$ (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:  \( 22137831.685215954 \) (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

\(\Q(\sqrt{113}) \), 4.4.1442897.1, 8.8.235260548044817.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ $16$

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
113Data not computed