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Properties

Label 16.6.81968949183...2631.2
Degree $16$
Signature $[6, 5]$
Discriminant $-\,71^{3}\cdot 73^{12}$
Root discriminant $55.54$
Ramified primes $71, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1251

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, -3872, -31268, 15254, -8119, -46025, 14824, 13538, -10578, 715, 2962, -740, -236, 121, -6, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 6*x^14 + 121*x^13 - 236*x^12 - 740*x^11 + 2962*x^10 + 715*x^9 - 10578*x^8 + 13538*x^7 + 14824*x^6 - 46025*x^5 - 8119*x^4 + 15254*x^3 - 31268*x^2 - 3872*x + 4096)
 
gp: K = bnfinit(x^16 - 6*x^15 - 6*x^14 + 121*x^13 - 236*x^12 - 740*x^11 + 2962*x^10 + 715*x^9 - 10578*x^8 + 13538*x^7 + 14824*x^6 - 46025*x^5 - 8119*x^4 + 15254*x^3 - 31268*x^2 - 3872*x + 4096, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 6 x^{14} + 121 x^{13} - 236 x^{12} - 740 x^{11} + 2962 x^{10} + 715 x^{9} - 10578 x^{8} + 13538 x^{7} + 14824 x^{6} - 46025 x^{5} - 8119 x^{4} + 15254 x^{3} - 31268 x^{2} - 3872 x + 4096 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-8196894918367374985699192631=-\,71^{3}\cdot 73^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.54$
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:  $71, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{13} + \frac{1}{16} a^{12} - \frac{1}{4} a^{9} - \frac{1}{8} a^{7} - \frac{3}{16} a^{6} + \frac{3}{16} a^{5} + \frac{1}{8} a^{4} - \frac{5}{16} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{12} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} + \frac{3}{16} a^{7} + \frac{3}{8} a^{6} + \frac{3}{16} a^{5} - \frac{3}{16} a^{4} - \frac{5}{16} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{4027639033028983612928227904096} a^{15} + \frac{11583788251282586161088146923}{503454879128622951616028488012} a^{14} - \frac{3898990412712235366326247784}{125863719782155737904007122003} a^{13} + \frac{132739717026913662255763413977}{4027639033028983612928227904096} a^{12} - \frac{88867578080037191150927454007}{1006909758257245903232056976024} a^{11} + \frac{81896432290292002632773664889}{1006909758257245903232056976024} a^{10} + \frac{423848700848198495431702050837}{2013819516514491806464113952048} a^{9} + \frac{801877217894909602137377939351}{4027639033028983612928227904096} a^{8} + \frac{111477145630721120450315510937}{1006909758257245903232056976024} a^{7} + \frac{435767493067461175492452162279}{1006909758257245903232056976024} a^{6} - \frac{149252423406371613717908787893}{1006909758257245903232056976024} a^{5} - \frac{484811480648234462556274669167}{4027639033028983612928227904096} a^{4} + \frac{587147222610931826256356384661}{4027639033028983612928227904096} a^{3} - \frac{510637111141544384071450674581}{2013819516514491806464113952048} a^{2} + \frac{433134584782494673142780086119}{1006909758257245903232056976024} a - \frac{29967935943371607505297872109}{125863719782155737904007122003}$

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:  $10$
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:  \( 793198315.417 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1251:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1251
Character table for t16n1251 is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.6.10744730066519.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ $16$

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
71Data not computed
$73$73.8.6.1$x^{8} - 14527 x^{4} + 78021889$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
73.8.6.1$x^{8} - 14527 x^{4} + 78021889$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$