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Properties

Label 16.4.17521464151...2529.1
Degree $16$
Signature $[4, 6]$
Discriminant $17^{14}\cdot 101^{4}$
Root discriminant $37.82$
Ramified primes $17, 101$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4861, 454, -2525, 13120, -10934, 12868, -7630, 2736, -1553, -108, -28, 22, 56, -28, 13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 13*x^14 - 28*x^13 + 56*x^12 + 22*x^11 - 28*x^10 - 108*x^9 - 1553*x^8 + 2736*x^7 - 7630*x^6 + 12868*x^5 - 10934*x^4 + 13120*x^3 - 2525*x^2 + 454*x + 4861)
 
gp: K = bnfinit(x^16 - 2*x^15 + 13*x^14 - 28*x^13 + 56*x^12 + 22*x^11 - 28*x^10 - 108*x^9 - 1553*x^8 + 2736*x^7 - 7630*x^6 + 12868*x^5 - 10934*x^4 + 13120*x^3 - 2525*x^2 + 454*x + 4861, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 13 x^{14} - 28 x^{13} + 56 x^{12} + 22 x^{11} - 28 x^{10} - 108 x^{9} - 1553 x^{8} + 2736 x^{7} - 7630 x^{6} + 12868 x^{5} - 10934 x^{4} + 13120 x^{3} - 2525 x^{2} + 454 x + 4861 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17521464151279710991512529=17^{14}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.82$
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:  $17, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} + \frac{1}{16} a^{12} - \frac{3}{16} a^{10} + \frac{3}{16} a^{9} + \frac{3}{16} a^{7} + \frac{1}{8} a^{6} - \frac{7}{16} a^{5} - \frac{3}{16} a^{4} + \frac{3}{8} a^{3} - \frac{7}{16} a^{2} - \frac{1}{16} a - \frac{1}{16}$, $\frac{1}{1565282737543708644780574135504} a^{15} + \frac{46689183819288506101851092093}{1565282737543708644780574135504} a^{14} - \frac{152245190253185945239162320405}{1565282737543708644780574135504} a^{13} + \frac{7539405473400684348391228827}{782641368771854322390287067752} a^{12} + \frac{373933803786971901202465468253}{1565282737543708644780574135504} a^{11} - \frac{292032200761993920091291130199}{1565282737543708644780574135504} a^{10} - \frac{101654199239669028383375755287}{782641368771854322390287067752} a^{9} + \frac{13890421155380917756504994427}{1565282737543708644780574135504} a^{8} + \frac{62633547330484126516934762549}{391320684385927161195143533876} a^{7} + \frac{5679256653552310337416076381}{1565282737543708644780574135504} a^{6} + \frac{600393580522295590690402242323}{1565282737543708644780574135504} a^{5} + \frac{108471978962100658193052557057}{391320684385927161195143533876} a^{4} - \frac{96887901824156620240425882739}{1565282737543708644780574135504} a^{3} - \frac{191070625850253530472538066771}{1565282737543708644780574135504} a^{2} + \frac{647992740600332261945986532697}{1565282737543708644780574135504} a - \frac{57847922848466462192961781411}{782641368771854322390287067752}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 943881.756869 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
101Data not computed