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Properties

Label 16.4.48763989276...3125.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{15}\cdot 41^{6}\cdot 18341^{2}$
Root discriminant $62.09$
Ramified primes $5, 41, 18341$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1113

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![35181701, 35305612, 6343430, -2753750, -9809590, -1161819, 2562842, 81435, -285830, 22360, 13312, -3201, 95, 140, -35, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 35*x^14 + 140*x^13 + 95*x^12 - 3201*x^11 + 13312*x^10 + 22360*x^9 - 285830*x^8 + 81435*x^7 + 2562842*x^6 - 1161819*x^5 - 9809590*x^4 - 2753750*x^3 + 6343430*x^2 + 35305612*x + 35181701)
 
gp: K = bnfinit(x^16 - 2*x^15 - 35*x^14 + 140*x^13 + 95*x^12 - 3201*x^11 + 13312*x^10 + 22360*x^9 - 285830*x^8 + 81435*x^7 + 2562842*x^6 - 1161819*x^5 - 9809590*x^4 - 2753750*x^3 + 6343430*x^2 + 35305612*x + 35181701, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 35 x^{14} + 140 x^{13} + 95 x^{12} - 3201 x^{11} + 13312 x^{10} + 22360 x^{9} - 285830 x^{8} + 81435 x^{7} + 2562842 x^{6} - 1161819 x^{5} - 9809590 x^{4} - 2753750 x^{3} + 6343430 x^{2} + 35305612 x + 35181701 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(48763989276665152618408203125=5^{15}\cdot 41^{6}\cdot 18341^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.09$
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:  $5, 41, 18341$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{41} a^{12} - \frac{17}{41} a^{11} + \frac{17}{41} a^{10} - \frac{12}{41} a^{9} + \frac{8}{41} a^{8} + \frac{14}{41} a^{7} - \frac{11}{41} a^{6} + \frac{10}{41} a^{5} - \frac{7}{41} a^{4} + \frac{13}{41} a^{2} + \frac{18}{41} a - \frac{20}{41}$, $\frac{1}{451} a^{13} + \frac{3}{451} a^{12} - \frac{36}{451} a^{11} - \frac{2}{11} a^{10} - \frac{150}{451} a^{9} - \frac{72}{451} a^{8} - \frac{100}{451} a^{7} + \frac{200}{451} a^{6} + \frac{193}{451} a^{5} - \frac{17}{451} a^{4} - \frac{151}{451} a^{3} - \frac{50}{451} a^{2} + \frac{217}{451} a + \frac{133}{451}$, $\frac{1}{451} a^{14} - \frac{1}{451} a^{12} + \frac{180}{451} a^{11} - \frac{58}{451} a^{10} - \frac{150}{451} a^{9} + \frac{17}{451} a^{8} + \frac{214}{451} a^{7} + \frac{1}{41} a^{6} - \frac{156}{451} a^{5} + \frac{43}{451} a^{4} - \frac{48}{451} a^{3} + \frac{37}{451} a^{2} - \frac{177}{451} a + \frac{74}{451}$, $\frac{1}{96369274152460342808809446769170520420045337731} a^{15} - \frac{1047429188840985139856415706826315300317348}{2350470101279520556312425530955378546830374091} a^{14} + \frac{73885030168100454158903996871350969530965422}{96369274152460342808809446769170520420045337731} a^{13} + \frac{430252114499816687868967158102069337457488455}{96369274152460342808809446769170520420045337731} a^{12} - \frac{1740968234593827830961033302656860929957774600}{8760843104769122073528131524470047310913212521} a^{11} + \frac{47332795995969008347663161773369067116880545903}{96369274152460342808809446769170520420045337731} a^{10} - \frac{37911257024389313571456297849691808351042635233}{96369274152460342808809446769170520420045337731} a^{9} + \frac{14704270466334110952122293919470750662894559582}{96369274152460342808809446769170520420045337731} a^{8} + \frac{5022325916020578807141603740670341312794682773}{96369274152460342808809446769170520420045337731} a^{7} - \frac{34507051783698421674356220998710279058710821584}{96369274152460342808809446769170520420045337731} a^{6} + \frac{8383105543488629125551097134146442766239319921}{96369274152460342808809446769170520420045337731} a^{5} - \frac{48180794563456065822887104869523627483366758751}{96369274152460342808809446769170520420045337731} a^{4} + \frac{44354135776139295309570603458446565348169508314}{96369274152460342808809446769170520420045337731} a^{3} + \frac{1282316192278448926989019666759742664128238328}{8760843104769122073528131524470047310913212521} a^{2} - \frac{32958053404234444865931553679924367359440257416}{96369274152460342808809446769170520420045337731} a - \frac{35747644249411588400067319688142249172956867495}{96369274152460342808809446769170520420045337731}$

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:  $9$
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:  \( 147276086.207 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1113:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 49 conjugacy class representatives for t16n1113
Character table for t16n1113 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.5125.1, 8.4.131328125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ $16$ R $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
5Data not computed
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
18341Data not computed