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Properties

Label 16.4.82524197056...5625.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 11^{10}\cdot 19^{4}$
Root discriminant $31.25$
Ramified primes $5, 11, 19$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T646)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![691, -2323, 2747, 4713, -11178, 9035, -7552, 6949, -3585, 1306, -567, 160, -53, 17, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 + 17*x^13 - 53*x^12 + 160*x^11 - 567*x^10 + 1306*x^9 - 3585*x^8 + 6949*x^7 - 7552*x^6 + 9035*x^5 - 11178*x^4 + 4713*x^3 + 2747*x^2 - 2323*x + 691)
 
gp: K = bnfinit(x^16 - 2*x^15 + 2*x^14 + 17*x^13 - 53*x^12 + 160*x^11 - 567*x^10 + 1306*x^9 - 3585*x^8 + 6949*x^7 - 7552*x^6 + 9035*x^5 - 11178*x^4 + 4713*x^3 + 2747*x^2 - 2323*x + 691, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 2 x^{14} + 17 x^{13} - 53 x^{12} + 160 x^{11} - 567 x^{10} + 1306 x^{9} - 3585 x^{8} + 6949 x^{7} - 7552 x^{6} + 9035 x^{5} - 11178 x^{4} + 4713 x^{3} + 2747 x^{2} - 2323 x + 691 \)

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:  \(825241970563213134765625=5^{12}\cdot 11^{10}\cdot 19^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.25$
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:  $5, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{20} a^{12} - \frac{1}{4} a^{11} - \frac{3}{20} a^{10} - \frac{1}{2} a^{9} + \frac{2}{5} a^{8} - \frac{1}{20} a^{7} + \frac{3}{20} a^{6} - \frac{1}{20} a^{5} + \frac{1}{10} a^{4} + \frac{1}{5} a^{3} - \frac{1}{4} a^{2} + \frac{7}{20} a - \frac{1}{20}$, $\frac{1}{20} a^{13} + \frac{1}{10} a^{11} - \frac{1}{4} a^{10} - \frac{1}{10} a^{9} - \frac{1}{20} a^{8} - \frac{1}{10} a^{7} + \frac{1}{5} a^{6} - \frac{3}{20} a^{5} - \frac{3}{10} a^{4} - \frac{1}{4} a^{3} + \frac{1}{10} a^{2} + \frac{1}{5} a - \frac{1}{4}$, $\frac{1}{380} a^{14} - \frac{1}{190} a^{13} + \frac{3}{380} a^{12} - \frac{1}{95} a^{11} + \frac{9}{76} a^{10} - \frac{167}{380} a^{9} + \frac{12}{95} a^{8} - \frac{153}{380} a^{7} - \frac{49}{190} a^{6} - \frac{101}{380} a^{5} - \frac{31}{380} a^{4} + \frac{39}{95} a^{3} + \frac{23}{76} a^{2} + \frac{6}{95} a + \frac{69}{380}$, $\frac{1}{38681589978927873031741220} a^{15} - \frac{3177551192513466987405}{3868158997892787303174122} a^{14} - \frac{67097002162893648647806}{9670397494731968257935305} a^{13} - \frac{462747673893245347810669}{19340794989463936515870610} a^{12} - \frac{6030490578881973315820779}{38681589978927873031741220} a^{11} + \frac{4021601758204216284759319}{19340794989463936515870610} a^{10} - \frac{10817577195256393497185}{3868158997892787303174122} a^{9} + \frac{1016180489043231124724573}{19340794989463936515870610} a^{8} + \frac{5484151600666719600878891}{19340794989463936515870610} a^{7} + \frac{654192494333630698856059}{2035873156785677527986380} a^{6} + \frac{2836914631477518324206}{508968289196419381996595} a^{5} - \frac{2569515131186518353968959}{19340794989463936515870610} a^{4} + \frac{1861636003650708034134621}{19340794989463936515870610} a^{3} + \frac{536309343812912357262917}{9670397494731968257935305} a^{2} + \frac{3865034430950168967504859}{7736317995785574606348244} a - \frac{8407243796131769715702447}{38681589978927873031741220}$

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:  \( 252270.619051 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T646):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 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

\(\Q(\sqrt{5}) \), 4.4.15125.1, 4.2.1375.1, 4.2.275.1, 8.4.228765625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$