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Properties

Label 16.4.58504429015...8125.3
Degree $16$
Signature $[4, 6]$
Discriminant $5^{15}\cdot 61^{8}$
Root discriminant $35.31$
Ramified primes $5, 61$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4:D_4.D_4$ (as 16T681)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![196, 2282, -7935, 7995, -9185, 3671, -903, -2690, 2470, -2185, 1032, -436, 120, -25, 10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 10*x^14 - 25*x^13 + 120*x^12 - 436*x^11 + 1032*x^10 - 2185*x^9 + 2470*x^8 - 2690*x^7 - 903*x^6 + 3671*x^5 - 9185*x^4 + 7995*x^3 - 7935*x^2 + 2282*x + 196)
 
gp: K = bnfinit(x^16 - 2*x^15 + 10*x^14 - 25*x^13 + 120*x^12 - 436*x^11 + 1032*x^10 - 2185*x^9 + 2470*x^8 - 2690*x^7 - 903*x^6 + 3671*x^5 - 9185*x^4 + 7995*x^3 - 7935*x^2 + 2282*x + 196, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 10 x^{14} - 25 x^{13} + 120 x^{12} - 436 x^{11} + 1032 x^{10} - 2185 x^{9} + 2470 x^{8} - 2690 x^{7} - 903 x^{6} + 3671 x^{5} - 9185 x^{4} + 7995 x^{3} - 7935 x^{2} + 2282 x + 196 \)

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:  \(5850442901528350830078125=5^{15}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.31$
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:  $5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{188743959849643337667848488} a^{15} + \frac{1422063780352333177118239}{188743959849643337667848488} a^{14} - \frac{11594016121844773215164871}{188743959849643337667848488} a^{13} + \frac{280095684531953948255524}{23592994981205417208481061} a^{12} + \frac{4651636613835416043334395}{94371979924821668833924244} a^{11} + \frac{686501857793863398638951}{23592994981205417208481061} a^{10} + \frac{12617550096045733102287049}{94371979924821668833924244} a^{9} + \frac{85665023124816243211991677}{188743959849643337667848488} a^{8} - \frac{14187049091654915207552059}{188743959849643337667848488} a^{7} - \frac{21130090890052242018614521}{188743959849643337667848488} a^{6} - \frac{3637047400018996608808539}{13481711417831666976274892} a^{5} - \frac{84742631490408950313128255}{188743959849643337667848488} a^{4} - \frac{38947423252430012955153313}{94371979924821668833924244} a^{3} + \frac{24190343320319311286960579}{188743959849643337667848488} a^{2} + \frac{25290183216970088423601439}{94371979924821668833924244} a - \frac{1858813289830732725049209}{6740855708915833488137446}$

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:  \( 945436.753486 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:D_4.D_4$ (as 16T681):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 19 conjugacy class representatives for $C_4:D_4.D_4$
Character table for $C_4:D_4.D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.7625.1, 8.4.290703125.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.8.6.2$x^{8} + 183 x^{4} + 14884$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$