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Properties

Label 16.0.14309137159...4256.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 31^{8}$
Root discriminant $15.75$
Ramified primes $2, 31$
Class number $3$
Class group $[3]$
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![1, -14, 63, -78, 137, -204, 126, -128, 218, -72, 22, -28, 33, -10, -1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - x^14 - 10*x^13 + 33*x^12 - 28*x^11 + 22*x^10 - 72*x^9 + 218*x^8 - 128*x^7 + 126*x^6 - 204*x^5 + 137*x^4 - 78*x^3 + 63*x^2 - 14*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 - x^14 - 10*x^13 + 33*x^12 - 28*x^11 + 22*x^10 - 72*x^9 + 218*x^8 - 128*x^7 + 126*x^6 - 204*x^5 + 137*x^4 - 78*x^3 + 63*x^2 - 14*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - x^{14} - 10 x^{13} + 33 x^{12} - 28 x^{11} + 22 x^{10} - 72 x^{9} + 218 x^{8} - 128 x^{7} + 126 x^{6} - 204 x^{5} + 137 x^{4} - 78 x^{3} + 63 x^{2} - 14 x + 1 \)

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:  \(14309137159611744256=2^{24}\cdot 31^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.75$
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:  $2, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{12} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a$, $\frac{1}{40} a^{14} + \frac{1}{10} a^{13} + \frac{1}{20} a^{12} - \frac{1}{20} a^{11} - \frac{1}{40} a^{10} + \frac{1}{5} a^{9} - \frac{9}{40} a^{8} + \frac{3}{20} a^{7} + \frac{3}{40} a^{6} + \frac{1}{10} a^{5} + \frac{7}{40} a^{4} - \frac{3}{20} a^{3} + \frac{7}{20} a^{2} - \frac{1}{5} a + \frac{1}{40}$, $\frac{1}{518088289360} a^{15} - \frac{2629624011}{518088289360} a^{14} + \frac{28212444301}{259044144680} a^{13} + \frac{2672167128}{32380518085} a^{12} + \frac{62371937569}{518088289360} a^{11} + \frac{4375522323}{518088289360} a^{10} - \frac{90421689529}{518088289360} a^{9} - \frac{108040675399}{518088289360} a^{8} + \frac{7815348253}{518088289360} a^{7} + \frac{109919521579}{518088289360} a^{6} + \frac{103010981447}{518088289360} a^{5} + \frac{171008798709}{518088289360} a^{4} - \frac{13956057177}{64761036170} a^{3} - \frac{16913653359}{259044144680} a^{2} + \frac{223221136121}{518088289360} a + \frac{1339653601}{103617657872}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( 599.577705049 \)
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

\(\Q(\sqrt{-62}) \), \(\Q(\sqrt{-31}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{-31})\), 4.0.7688.1 x2, 4.2.1984.1 x2, 8.0.3782742016.3, 8.0.472842752.1 x4, 8.2.122023936.2 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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$31$31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$