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Properties

Label 16.0.35847274805...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 59^{8}$
Root discriminant $25.68$
Ramified primes $5, 59$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9801, -8613, 1413, -633, 349, -1898, 2407, -662, 148, -202, 97, -104, 50, 6, -1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - x^14 + 6*x^13 + 50*x^12 - 104*x^11 + 97*x^10 - 202*x^9 + 148*x^8 - 662*x^7 + 2407*x^6 - 1898*x^5 + 349*x^4 - 633*x^3 + 1413*x^2 - 8613*x + 9801)
 
gp: K = bnfinit(x^16 - 4*x^15 - x^14 + 6*x^13 + 50*x^12 - 104*x^11 + 97*x^10 - 202*x^9 + 148*x^8 - 662*x^7 + 2407*x^6 - 1898*x^5 + 349*x^4 - 633*x^3 + 1413*x^2 - 8613*x + 9801, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - x^{14} + 6 x^{13} + 50 x^{12} - 104 x^{11} + 97 x^{10} - 202 x^{9} + 148 x^{8} - 662 x^{7} + 2407 x^{6} - 1898 x^{5} + 349 x^{4} - 633 x^{3} + 1413 x^{2} - 8613 x + 9801 \)

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:  \(35847274805742431640625=5^{12}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.68$
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:  $5, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{15} a^{12} + \frac{2}{15} a^{11} - \frac{4}{15} a^{8} - \frac{4}{15} a^{6} + \frac{1}{3} a^{5} + \frac{1}{15} a^{4} - \frac{1}{3} a^{3} - \frac{7}{15} a + \frac{2}{5}$, $\frac{1}{15} a^{13} + \frac{1}{15} a^{11} + \frac{1}{15} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{7}{15} a^{6} + \frac{1}{15} a^{5} + \frac{1}{5} a^{4} - \frac{7}{15} a^{2} + \frac{1}{3} a + \frac{1}{5}$, $\frac{1}{225} a^{14} - \frac{1}{225} a^{13} + \frac{1}{45} a^{12} + \frac{4}{75} a^{11} - \frac{19}{225} a^{10} + \frac{22}{225} a^{9} + \frac{67}{225} a^{8} - \frac{28}{225} a^{7} + \frac{67}{225} a^{6} + \frac{37}{225} a^{5} + \frac{46}{225} a^{4} - \frac{2}{225} a^{3} + \frac{82}{225} a^{2} - \frac{7}{15} a + \frac{4}{25}$, $\frac{1}{5595646610705092425} a^{15} - \frac{196600383325087}{223825864428203697} a^{14} + \frac{156083793575536529}{5595646610705092425} a^{13} - \frac{54713305090044346}{1865215536901697475} a^{12} - \frac{918996236754792817}{5595646610705092425} a^{11} + \frac{420210886701997903}{5595646610705092425} a^{10} + \frac{247168335832224739}{5595646610705092425} a^{9} - \frac{2196058554323287141}{5595646610705092425} a^{8} + \frac{1025801698002959839}{5595646610705092425} a^{7} + \frac{462160012686805849}{5595646610705092425} a^{6} - \frac{1024042050800339342}{5595646610705092425} a^{5} - \frac{1660310162004614786}{5595646610705092425} a^{4} - \frac{475454201623466719}{1119129322141018485} a^{3} - \frac{131831994451533466}{1865215536901697475} a^{2} - \frac{260704269429042326}{621738512300565825} a - \frac{8608893617689527}{18840560978805025}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{366790754980531}{621738512300565825} a^{15} + \frac{486048446398028}{621738512300565825} a^{14} + \frac{1886842967269988}{621738512300565825} a^{13} + \frac{2340930729350828}{621738512300565825} a^{12} - \frac{11546177485342402}{621738512300565825} a^{11} + \frac{206819006209075}{24869540492022633} a^{10} - \frac{19569587271637003}{621738512300565825} a^{9} + \frac{365036486912459}{207246170766855275} a^{8} - \frac{1025740466923476}{207246170766855275} a^{7} + \frac{211515478235730112}{621738512300565825} a^{6} - \frac{197305122760544872}{621738512300565825} a^{5} + \frac{6145699782235303}{207246170766855275} a^{4} + \frac{35429103379506139}{621738512300565825} a^{3} - \frac{206869257287137796}{621738512300565825} a^{2} - \frac{714093267047044081}{621738512300565825} a + \frac{45133896118567344}{18840560978805025} \) (order $10$)
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:  \( 226000.220646 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.2.1475.1, 4.2.7375.1, 8.4.37866753125.1, 8.0.37866753125.1, 8.0.54390625.1

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
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ 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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$59$59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.8.6.1$x^{8} - 59 x^{4} + 55696$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$