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Properties

Label 16.0.342974209600000000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 5^{8}\cdot 11^{8}$
Root discriminant $12.47$
Ramified primes $2, 5, 11$
Class number $1$
Class group Trivial
Galois group $D_8:C_2$ (as 16T47)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 28, -52, 49, -17, 8, -28, 20, 8, -3, -7, -6, 9, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 + 9*x^13 - 6*x^12 - 7*x^11 - 3*x^10 + 8*x^9 + 20*x^8 - 28*x^7 + 8*x^6 - 17*x^5 + 49*x^4 - 52*x^3 + 28*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 2*x^14 + 9*x^13 - 6*x^12 - 7*x^11 - 3*x^10 + 8*x^9 + 20*x^8 - 28*x^7 + 8*x^6 - 17*x^5 + 49*x^4 - 52*x^3 + 28*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 2 x^{14} + 9 x^{13} - 6 x^{12} - 7 x^{11} - 3 x^{10} + 8 x^{9} + 20 x^{8} - 28 x^{7} + 8 x^{6} - 17 x^{5} + 49 x^{4} - 52 x^{3} + 28 x^{2} - 8 x + 1 \)

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:  \(342974209600000000=2^{12}\cdot 5^{8}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.47$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{71} a^{14} - \frac{19}{71} a^{13} - \frac{34}{71} a^{12} + \frac{15}{71} a^{11} + \frac{33}{71} a^{10} + \frac{8}{71} a^{9} + \frac{5}{71} a^{8} - \frac{8}{71} a^{7} + \frac{26}{71} a^{6} + \frac{20}{71} a^{5} + \frac{24}{71} a^{4} + \frac{19}{71} a^{3} + \frac{12}{71} a^{2} - \frac{12}{71} a - \frac{23}{71}$, $\frac{1}{83851} a^{15} - \frac{494}{83851} a^{14} + \frac{15310}{83851} a^{13} - \frac{7265}{83851} a^{12} + \frac{33378}{83851} a^{11} + \frac{32329}{83851} a^{10} - \frac{13451}{83851} a^{9} - \frac{2525}{83851} a^{8} + \frac{19659}{83851} a^{7} - \frac{20779}{83851} a^{6} + \frac{9765}{83851} a^{5} - \frac{12446}{83851} a^{4} - \frac{14267}{83851} a^{3} - \frac{742}{83851} a^{2} - \frac{9588}{83851} a + \frac{40248}{83851}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( 88.5494213757 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8:C_2$ (as 16T47):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-55}) \), 4.0.605.1 x2, 4.2.275.1 x2, \(\Q(\sqrt{5}, \sqrt{-11})\), 8.0.9150625.1

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 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$2$2.8.12.5$x^{8} + 6 x^{6} + 8 x^{5} + 80$$2$$4$$12$$C_8$$[3]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$