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Properties

Label 16.0.14874737901...6201.1
Degree $16$
Signature $[0, 8]$
Discriminant $47^{6}\cdot 53^{14}$
Root discriminant $136.70$
Ramified primes $47, 53$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![26722273, 53884137, 57198013, 29813275, 7439752, -2904858, -1221602, -735816, 59155, 9204, 20046, -274, 148, -218, -1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - x^14 - 218*x^13 + 148*x^12 - 274*x^11 + 20046*x^10 + 9204*x^9 + 59155*x^8 - 735816*x^7 - 1221602*x^6 - 2904858*x^5 + 7439752*x^4 + 29813275*x^3 + 57198013*x^2 + 53884137*x + 26722273)
 
gp: K = bnfinit(x^16 - 2*x^15 - x^14 - 218*x^13 + 148*x^12 - 274*x^11 + 20046*x^10 + 9204*x^9 + 59155*x^8 - 735816*x^7 - 1221602*x^6 - 2904858*x^5 + 7439752*x^4 + 29813275*x^3 + 57198013*x^2 + 53884137*x + 26722273, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - x^{14} - 218 x^{13} + 148 x^{12} - 274 x^{11} + 20046 x^{10} + 9204 x^{9} + 59155 x^{8} - 735816 x^{7} - 1221602 x^{6} - 2904858 x^{5} + 7439752 x^{4} + 29813275 x^{3} + 57198013 x^{2} + 53884137 x + 26722273 \)

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:  \(14874737901162615182149330162516201=47^{6}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $136.70$
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:  $47, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} + \frac{3}{11} a^{12} + \frac{1}{11} a^{11} + \frac{4}{11} a^{7} - \frac{5}{11} a^{6} - \frac{3}{11} a^{5} - \frac{2}{11} a^{4} + \frac{4}{11} a^{3} + \frac{5}{11} a^{2} - \frac{2}{11} a - \frac{5}{11}$, $\frac{1}{517} a^{14} - \frac{2}{517} a^{13} + \frac{140}{517} a^{12} + \frac{17}{517} a^{11} + \frac{22}{47} a^{10} + \frac{5}{47} a^{9} - \frac{117}{517} a^{8} - \frac{102}{517} a^{7} - \frac{23}{47} a^{6} - \frac{31}{517} a^{5} + \frac{69}{517} a^{4} - \frac{70}{517} a^{3} + \frac{28}{517} a^{2} - \frac{2}{11} a + \frac{1}{11}$, $\frac{1}{12962103220039710456454986827742421171674798094769} a^{15} - \frac{9432123841902971415464503391429504732649441852}{12962103220039710456454986827742421171674798094769} a^{14} - \frac{339711804214453593428307540692981072000675093229}{12962103220039710456454986827742421171674798094769} a^{13} + \frac{5868799098223183028005043181705931159986599454405}{12962103220039710456454986827742421171674798094769} a^{12} + \frac{4250740574731059449729273380492451564720635765993}{12962103220039710456454986827742421171674798094769} a^{11} - \frac{405272832590494509373401065996586620435225008408}{1178373020003610041495907893431129197424981644979} a^{10} - \frac{6288971181195590647621588527232868787415316280101}{12962103220039710456454986827742421171674798094769} a^{9} + \frac{6403052011618234170334615640275590219240712018747}{12962103220039710456454986827742421171674798094769} a^{8} - \frac{965976199673236573570791725054785706508316692490}{12962103220039710456454986827742421171674798094769} a^{7} + \frac{48510641994454829959464842360566816468053288208}{997084863079977727419614371364801628590369084213} a^{6} - \frac{3249652944599882421743612587766710146945546364735}{12962103220039710456454986827742421171674798094769} a^{5} + \frac{421565893015885650392588159870182455728491975621}{12962103220039710456454986827742421171674798094769} a^{4} + \frac{1014332257007724124984150533222020830997877536779}{12962103220039710456454986827742421171674798094769} a^{3} + \frac{12436192924724965568162662429288770559879342867}{997084863079977727419614371364801628590369084213} a^{2} - \frac{135444582766883599333443499157212956931688039864}{275789430213610860775638017611540875993080810527} a - \frac{94333154519369219329383784972596298554861047846}{275789430213610860775638017611540875993080810527}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 5340169342.87 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{53}) \), 4.2.132023.1, 4.0.148877.1, 4.2.6997219.1, 8.0.2594936907899933.1 x2, 8.0.48961073733961.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ R R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
47Data not computed
53Data not computed