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Properties

Label 16.0.47267635305...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{4}\cdot 229^{8}$
Root discriminant $22.63$
Ramified primes $5, 229$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $\GL(2,Z/4)$ (as 16T193)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![51, -164, 137, 28, 204, -368, 453, -318, 283, -308, 149, -92, 50, -16, 11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 11*x^14 - 16*x^13 + 50*x^12 - 92*x^11 + 149*x^10 - 308*x^9 + 283*x^8 - 318*x^7 + 453*x^6 - 368*x^5 + 204*x^4 + 28*x^3 + 137*x^2 - 164*x + 51)
 
gp: K = bnfinit(x^16 - 2*x^15 + 11*x^14 - 16*x^13 + 50*x^12 - 92*x^11 + 149*x^10 - 308*x^9 + 283*x^8 - 318*x^7 + 453*x^6 - 368*x^5 + 204*x^4 + 28*x^3 + 137*x^2 - 164*x + 51, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 11 x^{14} - 16 x^{13} + 50 x^{12} - 92 x^{11} + 149 x^{10} - 308 x^{9} + 283 x^{8} - 318 x^{7} + 453 x^{6} - 368 x^{5} + 204 x^{4} + 28 x^{3} + 137 x^{2} - 164 x + 51 \)

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:  \(4726763530575017100625=5^{4}\cdot 229^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.63$
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:  $5, 229$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{14} a^{13} - \frac{1}{14} a^{12} - \frac{1}{7} a^{11} - \frac{1}{7} a^{9} - \frac{1}{14} a^{8} - \frac{3}{14} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{2}{7} a^{4} + \frac{1}{14} a^{3} + \frac{1}{14} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{31262} a^{14} + \frac{57}{31262} a^{13} + \frac{5603}{31262} a^{12} + \frac{4315}{31262} a^{11} + \frac{4457}{31262} a^{10} + \frac{1107}{15631} a^{9} - \frac{468}{15631} a^{8} - \frac{13029}{31262} a^{7} - \frac{4965}{15631} a^{6} - \frac{46}{203} a^{5} + \frac{10}{77} a^{4} + \frac{1781}{31262} a^{3} + \frac{5444}{15631} a^{2} + \frac{562}{15631} a + \frac{13969}{31262}$, $\frac{1}{68619809548598} a^{15} - \frac{1839615}{215109120842} a^{14} + \frac{83235449923}{6238164504418} a^{13} + \frac{251862692195}{2366200329262} a^{12} + \frac{1797110028954}{34309904774299} a^{11} + \frac{14669626071231}{68619809548598} a^{10} + \frac{15086285619299}{68619809548598} a^{9} + \frac{1694537787049}{68619809548598} a^{8} + \frac{34117816932443}{68619809548598} a^{7} + \frac{13061061592631}{34309904774299} a^{6} - \frac{2191315964511}{9802829935514} a^{5} + \frac{190894708007}{1595809524386} a^{4} + \frac{1642690146661}{6238164504418} a^{3} - \frac{25220682091049}{68619809548598} a^{2} + \frac{16973967050494}{34309904774299} a + \frac{9965336653262}{34309904774299}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( -1 \) (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:  \( 17934.8862387 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\GL(2,Z/4)$ (as 16T193):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 14 conjugacy class representatives for $\GL(2,Z/4)$
Character table for $\GL(2,Z/4)$

Intermediate fields

\(\Q(\sqrt{229}) \), 4.0.229.1, 4.4.262205.1, 8.0.2750058481.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: 12.4.9840053002328125.1, 12.0.9840053002328125.1, 12.0.9840053002328125.2, 12.4.18026977100265125.2
Degree 16 sibling: 16.0.56334303438328515625.1
Degree 24 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
229Data not computed