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Properties

Label 16.8.71085478368...1921.1
Degree $16$
Signature $[8, 4]$
Discriminant $37^{4}\cdot 41^{14}$
Root discriminant $63.57$
Ramified primes $37, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1231, -17, -11990, 39343, -58720, 47452, -12455, -11666, 13980, -7273, 942, 784, -525, 154, -2, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 2*x^14 + 154*x^13 - 525*x^12 + 784*x^11 + 942*x^10 - 7273*x^9 + 13980*x^8 - 11666*x^7 - 12455*x^6 + 47452*x^5 - 58720*x^4 + 39343*x^3 - 11990*x^2 - 17*x + 1231)
 
gp: K = bnfinit(x^16 - 8*x^15 - 2*x^14 + 154*x^13 - 525*x^12 + 784*x^11 + 942*x^10 - 7273*x^9 + 13980*x^8 - 11666*x^7 - 12455*x^6 + 47452*x^5 - 58720*x^4 + 39343*x^3 - 11990*x^2 - 17*x + 1231, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 2 x^{14} + 154 x^{13} - 525 x^{12} + 784 x^{11} + 942 x^{10} - 7273 x^{9} + 13980 x^{8} - 11666 x^{7} - 12455 x^{6} + 47452 x^{5} - 58720 x^{4} + 39343 x^{3} - 11990 x^{2} - 17 x + 1231 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(71085478368850138600216861921=37^{4}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.57$
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:  $37, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{37600937459374} a^{14} - \frac{7}{37600937459374} a^{13} + \frac{3732144593605}{18800468729687} a^{12} - \frac{7184797663795}{37600937459374} a^{11} + \frac{13456239519779}{37600937459374} a^{10} + \frac{2423135281144}{18800468729687} a^{9} + \frac{12996810204221}{37600937459374} a^{8} - \frac{9693888654575}{37600937459374} a^{7} + \frac{6707012567002}{18800468729687} a^{6} - \frac{8448885588255}{37600937459374} a^{5} - \frac{16493026713287}{37600937459374} a^{4} + \frac{5808178003807}{18800468729687} a^{3} + \frac{5439415505947}{37600937459374} a^{2} + \frac{10188129958229}{37600937459374} a + \frac{8328650811605}{18800468729687}$, $\frac{1}{256776801910065046} a^{15} + \frac{3407}{256776801910065046} a^{14} + \frac{20120233685346746}{128388400955032523} a^{13} + \frac{20700579430130647}{256776801910065046} a^{12} + \frac{128294766850219585}{256776801910065046} a^{11} + \frac{46284805220355777}{128388400955032523} a^{10} + \frac{86571110059209841}{256776801910065046} a^{9} + \frac{6760478689181919}{256776801910065046} a^{8} - \frac{21447683259240830}{128388400955032523} a^{7} - \frac{61638845399530117}{256776801910065046} a^{6} + \frac{47619318367350859}{256776801910065046} a^{5} + \frac{13908860126732717}{128388400955032523} a^{4} + \frac{59892983178643355}{256776801910065046} a^{3} - \frac{9281941990134847}{256776801910065046} a^{2} - \frac{14166520507635965}{128388400955032523} a + \frac{552918744693649}{128388400955032523}$

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:  $11$
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:  \( 336720308.906 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.266618600943089.1, 8.4.7205908133597.1, 8.4.175753856917.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ R R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$37$37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$