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Properties

Label 16.8.61103770654...6032.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 113^{8}\cdot 137$
Root discriminant $40.89$
Ramified primes $2, 113, 137$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1638

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![617, -2502, 1621, 1884, -3365, 872, 967, 482, -402, -582, 509, -116, -99, 60, -9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 9*x^14 + 60*x^13 - 99*x^12 - 116*x^11 + 509*x^10 - 582*x^9 - 402*x^8 + 482*x^7 + 967*x^6 + 872*x^5 - 3365*x^4 + 1884*x^3 + 1621*x^2 - 2502*x + 617)
 
gp: K = bnfinit(x^16 - 2*x^15 - 9*x^14 + 60*x^13 - 99*x^12 - 116*x^11 + 509*x^10 - 582*x^9 - 402*x^8 + 482*x^7 + 967*x^6 + 872*x^5 - 3365*x^4 + 1884*x^3 + 1621*x^2 - 2502*x + 617, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 9 x^{14} + 60 x^{13} - 99 x^{12} - 116 x^{11} + 509 x^{10} - 582 x^{9} - 402 x^{8} + 482 x^{7} + 967 x^{6} + 872 x^{5} - 3365 x^{4} + 1884 x^{3} + 1621 x^{2} - 2502 x + 617 \)

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:  \(61103770654221524409516032=2^{24}\cdot 113^{8}\cdot 137\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.89$
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:  $2, 113, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{3904} a^{14} + \frac{35}{976} a^{13} + \frac{99}{1952} a^{12} + \frac{225}{976} a^{11} - \frac{193}{3904} a^{10} + \frac{349}{1952} a^{9} + \frac{161}{1952} a^{8} + \frac{203}{976} a^{7} - \frac{3}{16} a^{6} - \frac{633}{1952} a^{5} - \frac{441}{3904} a^{4} - \frac{197}{976} a^{3} - \frac{235}{976} a^{2} - \frac{159}{488} a - \frac{47}{3904}$, $\frac{1}{83348652931891207971968} a^{15} - \frac{4535329482895195795}{83348652931891207971968} a^{14} - \frac{1251392114180689314831}{41674326465945603985984} a^{13} + \frac{4715242231723876804557}{41674326465945603985984} a^{12} + \frac{15147630035289177840787}{83348652931891207971968} a^{11} + \frac{32530888210093712071}{582857712810428027776} a^{10} + \frac{45693791787815932691}{341592839884800032672} a^{9} - \frac{6465491334306288701321}{41674326465945603985984} a^{8} - \frac{580200476547354362441}{5209290808243200498248} a^{7} + \frac{212647988907156011305}{41674326465945603985984} a^{6} - \frac{23992096607188944712971}{83348652931891207971968} a^{5} - \frac{34483297997599827098413}{83348652931891207971968} a^{4} - \frac{45267230976497871923}{473571891658472772568} a^{3} + \frac{1153633236310966976211}{20837163232972801992992} a^{2} - \frac{23794135114282008396759}{83348652931891207971968} a - \frac{6215412437114258048383}{83348652931891207971968}$

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:  $11$
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:  \( 7564009.62792 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1638:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1638 are not computed
Character table for t16n1638 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.7232.1, 8.8.5910106112.1

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 R $16$ $16$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$113$113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.4.2.2$x^{4} - 113 x^{2} + 127690$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
113.8.4.1$x^{8} + 127690 x^{4} - 1442897 x^{2} + 4076184025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
137Data not computed