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Properties

Label 16.0.75293116527...2401.1
Degree $16$
Signature $[0, 8]$
Discriminant $11^{8}\cdot 37^{8}$
Root discriminant $20.17$
Ramified primes $11, 37$
Class number $2$
Class group $[2]$
Galois group $D_{8}$ (as 16T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121, 121, 143, -715, 960, -1006, 583, 58, 3, -459, 358, 68, -163, 42, 14, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 14*x^14 + 42*x^13 - 163*x^12 + 68*x^11 + 358*x^10 - 459*x^9 + 3*x^8 + 58*x^7 + 583*x^6 - 1006*x^5 + 960*x^4 - 715*x^3 + 143*x^2 + 121*x + 121)
 
gp: K = bnfinit(x^16 - 8*x^15 + 14*x^14 + 42*x^13 - 163*x^12 + 68*x^11 + 358*x^10 - 459*x^9 + 3*x^8 + 58*x^7 + 583*x^6 - 1006*x^5 + 960*x^4 - 715*x^3 + 143*x^2 + 121*x + 121, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 14 x^{14} + 42 x^{13} - 163 x^{12} + 68 x^{11} + 358 x^{10} - 459 x^{9} + 3 x^{8} + 58 x^{7} + 583 x^{6} - 1006 x^{5} + 960 x^{4} - 715 x^{3} + 143 x^{2} + 121 x + 121 \)

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:  \(752931165277996622401=11^{8}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.17$
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:  $11, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{1265} a^{12} - \frac{6}{1265} a^{11} + \frac{309}{1265} a^{10} - \frac{45}{253} a^{9} + \frac{277}{1265} a^{8} + \frac{16}{115} a^{7} + \frac{124}{253} a^{6} + \frac{17}{115} a^{5} - \frac{298}{1265} a^{4} + \frac{97}{253} a^{3} - \frac{349}{1265} a^{2} + \frac{8}{115} a - \frac{4}{115}$, $\frac{1}{1265} a^{13} + \frac{273}{1265} a^{11} + \frac{364}{1265} a^{10} + \frac{192}{1265} a^{9} + \frac{573}{1265} a^{8} + \frac{411}{1265} a^{7} + \frac{112}{1265} a^{6} - \frac{441}{1265} a^{5} - \frac{38}{1265} a^{4} + \frac{31}{1265} a^{3} + \frac{524}{1265} a^{2} + \frac{44}{115} a - \frac{24}{115}$, $\frac{1}{37678025} a^{14} - \frac{1}{5382575} a^{13} + \frac{22}{148925} a^{12} - \frac{6661}{7535605} a^{11} + \frac{90013}{489325} a^{10} + \frac{302559}{3425275} a^{9} + \frac{263513}{5382575} a^{8} + \frac{9964698}{37678025} a^{7} + \frac{1069489}{7535605} a^{6} + \frac{2481387}{7535605} a^{5} - \frac{9659102}{37678025} a^{4} + \frac{9705807}{37678025} a^{3} - \frac{7756726}{37678025} a^{2} + \frac{72636}{489325} a - \frac{632804}{3425275}$, $\frac{1}{3353344225} a^{15} + \frac{1}{90630925} a^{14} - \frac{947862}{3353344225} a^{13} + \frac{1105149}{3353344225} a^{12} + \frac{116682771}{3353344225} a^{11} + \frac{1480838288}{3353344225} a^{10} + \frac{1610458582}{3353344225} a^{9} + \frac{66768447}{304849475} a^{8} - \frac{54419538}{3353344225} a^{7} + \frac{12726633}{134133769} a^{6} - \frac{458811242}{3353344225} a^{5} + \frac{824088954}{3353344225} a^{4} + \frac{52998216}{479049175} a^{3} + \frac{284778148}{3353344225} a^{2} + \frac{4076579}{304849475} a - \frac{94174571}{304849475}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 5099.43681828 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8$ (as 16T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-407}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{-11}, \sqrt{37})\), 4.0.4477.1 x2, 4.2.15059.1 x2, 8.0.27439591201.1, 8.0.741610573.1 x4, 8.2.2494508291.1 x4

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
$37$37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$