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Properties

Label 16.4.98340377973...9249.4
Degree $16$
Signature $[4, 6]$
Discriminant $13^{10}\cdot 61^{10}$
Root discriminant $64.87$
Ramified primes $13, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T875

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-231183, 744165, 188282, -471125, 324725, 29159, -31338, 48936, -7128, -6485, 4268, -259, -457, 113, 1, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + x^14 + 113*x^13 - 457*x^12 - 259*x^11 + 4268*x^10 - 6485*x^9 - 7128*x^8 + 48936*x^7 - 31338*x^6 + 29159*x^5 + 324725*x^4 - 471125*x^3 + 188282*x^2 + 744165*x - 231183)
 
gp: K = bnfinit(x^16 - 6*x^15 + x^14 + 113*x^13 - 457*x^12 - 259*x^11 + 4268*x^10 - 6485*x^9 - 7128*x^8 + 48936*x^7 - 31338*x^6 + 29159*x^5 + 324725*x^4 - 471125*x^3 + 188282*x^2 + 744165*x - 231183, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + x^{14} + 113 x^{13} - 457 x^{12} - 259 x^{11} + 4268 x^{10} - 6485 x^{9} - 7128 x^{8} + 48936 x^{7} - 31338 x^{6} + 29159 x^{5} + 324725 x^{4} - 471125 x^{3} + 188282 x^{2} + 744165 x - 231183 \)

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:  \(98340377973019428085802419249=13^{10}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.87$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{39} a^{14} - \frac{17}{39} a^{13} + \frac{5}{39} a^{12} + \frac{10}{39} a^{11} + \frac{17}{39} a^{9} - \frac{14}{39} a^{8} - \frac{4}{39} a^{7} + \frac{2}{39} a^{6} - \frac{1}{39} a^{5} + \frac{14}{39} a^{4} + \frac{16}{39} a^{3} + \frac{1}{13} a^{2} - \frac{2}{39} a + \frac{3}{13}$, $\frac{1}{39368818067385339329412065179552283413381377} a^{15} + \frac{150547445502888083643366170703700219126112}{39368818067385339329412065179552283413381377} a^{14} + \frac{5829968779013510767800320549516271803917018}{13122939355795113109804021726517427804460459} a^{13} + \frac{5051709104021547442353142691760642835010400}{13122939355795113109804021726517427804460459} a^{12} + \frac{14652826158040850417767866083825800216861372}{39368818067385339329412065179552283413381377} a^{11} - \frac{1518993019571531413359672456795372473355781}{39368818067385339329412065179552283413381377} a^{10} - \frac{40161729909682618052207143319611753796857}{279211475655215172549021738862072932009797} a^{9} - \frac{3279240057646243798207684934523826749623138}{13122939355795113109804021726517427804460459} a^{8} - \frac{7069361373634618264424825643931654796707652}{39368818067385339329412065179552283413381377} a^{7} + \frac{17563582820869674134534934298448785813563190}{39368818067385339329412065179552283413381377} a^{6} + \frac{8707166979506305851214810273628029829686684}{39368818067385339329412065179552283413381377} a^{5} - \frac{4398075654180162472305902403582530153319138}{13122939355795113109804021726517427804460459} a^{4} - \frac{253099540269673040315812756655142450909683}{3028370620568103025339389629196329493337029} a^{3} + \frac{3299297069994674024315551207809176861155678}{39368818067385339329412065179552283413381377} a^{2} + \frac{14669162668758860578378555224992551551089425}{39368818067385339329412065179552283413381377} a + \frac{5112679328688749332691089171252382130740410}{13122939355795113109804021726517427804460459}$

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:  \( 315557107.393 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T875:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n875
Character table for t16n875 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.10309.1, 8.4.395451064801.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61Data not computed