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Properties

Label 16.8.20402949088...8369.1
Degree $16$
Signature $[8, 4]$
Discriminant $17^{14}\cdot 59^{4}$
Root discriminant $33.06$
Ramified primes $17, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, 1216, 5600, 4528, -8065, -7778, 3, 2351, 1927, 465, -29, 16, -27, -14, -6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 6*x^14 - 14*x^13 - 27*x^12 + 16*x^11 - 29*x^10 + 465*x^9 + 1927*x^8 + 2351*x^7 + 3*x^6 - 7778*x^5 - 8065*x^4 + 4528*x^3 + 5600*x^2 + 1216*x + 256)
 
gp: K = bnfinit(x^16 - x^15 - 6*x^14 - 14*x^13 - 27*x^12 + 16*x^11 - 29*x^10 + 465*x^9 + 1927*x^8 + 2351*x^7 + 3*x^6 - 7778*x^5 - 8065*x^4 + 4528*x^3 + 5600*x^2 + 1216*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 6 x^{14} - 14 x^{13} - 27 x^{12} + 16 x^{11} - 29 x^{10} + 465 x^{9} + 1927 x^{8} + 2351 x^{7} + 3 x^{6} - 7778 x^{5} - 8065 x^{4} + 4528 x^{3} + 5600 x^{2} + 1216 x + 256 \)

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:  \(2040294908815649000428369=17^{14}\cdot 59^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.06$
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:  $17, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{208} a^{14} - \frac{21}{208} a^{13} + \frac{11}{104} a^{12} - \frac{11}{104} a^{11} + \frac{5}{208} a^{10} + \frac{3}{52} a^{9} + \frac{59}{208} a^{8} - \frac{11}{208} a^{7} - \frac{77}{208} a^{6} - \frac{1}{16} a^{5} + \frac{79}{208} a^{4} - \frac{51}{104} a^{3} + \frac{31}{208} a^{2} - \frac{25}{52} a - \frac{5}{13}$, $\frac{1}{26779767011226994301547712} a^{15} - \frac{27522884579040649184161}{26779767011226994301547712} a^{14} - \frac{325782416197596160903891}{13389883505613497150773856} a^{13} + \frac{1114058759184011648160105}{13389883505613497150773856} a^{12} + \frac{5448982520854547415332645}{26779767011226994301547712} a^{11} + \frac{108061404315029451302261}{1673735438201687143846732} a^{10} + \frac{2083899208129093521403427}{26779767011226994301547712} a^{9} - \frac{58805783543159498219045}{399698015092940213455936} a^{8} + \frac{4485083067018453600790439}{26779767011226994301547712} a^{7} + \frac{111125653183358670846503}{300896258553112295523008} a^{6} - \frac{9865491723020471980327261}{26779767011226994301547712} a^{5} + \frac{5574553009500262889497871}{13389883505613497150773856} a^{4} - \frac{5273318563205718142256097}{26779767011226994301547712} a^{3} - \frac{628798482318510156660653}{1673735438201687143846732} a^{2} - \frac{26263381434544669307796}{418433859550421785961683} a + \frac{13407677370696428423420}{418433859550421785961683}$

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:  \( 1322848.31436 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.2.289867.1, 4.4.4913.1, 4.2.17051.1, 8.4.1428388920713.1, \(\Q(\zeta_{17})^+\), 8.4.84022877689.1

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 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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
17Data not computed
$59$59.4.0.1$x^{4} - x + 14$$1$$4$$0$$C_4$$[\ ]^{4}$
59.4.0.1$x^{4} - x + 14$$1$$4$$0$$C_4$$[\ ]^{4}$
59.8.4.1$x^{8} + 97468 x^{4} - 205379 x^{2} + 2375002756$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$