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Properties

Label 16.0.63141896891...3121.2
Degree $16$
Signature $[0, 8]$
Discriminant $23^{4}\cdot 41^{12}$
Root discriminant $35.48$
Ramified primes $23, 41$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T158)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -200, 577, 1694, 2992, 3336, 3576, 3734, 2713, 1148, 478, 280, 102, 18, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 5*x^14 + 18*x^13 + 102*x^12 + 280*x^11 + 478*x^10 + 1148*x^9 + 2713*x^8 + 3734*x^7 + 3576*x^6 + 3336*x^5 + 2992*x^4 + 1694*x^3 + 577*x^2 - 200*x + 25)
 
gp: K = bnfinit(x^16 - 2*x^15 + 5*x^14 + 18*x^13 + 102*x^12 + 280*x^11 + 478*x^10 + 1148*x^9 + 2713*x^8 + 3734*x^7 + 3576*x^6 + 3336*x^5 + 2992*x^4 + 1694*x^3 + 577*x^2 - 200*x + 25, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 5 x^{14} + 18 x^{13} + 102 x^{12} + 280 x^{11} + 478 x^{10} + 1148 x^{9} + 2713 x^{8} + 3734 x^{7} + 3576 x^{6} + 3336 x^{5} + 2992 x^{4} + 1694 x^{3} + 577 x^{2} - 200 x + 25 \)

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:  \(6314189689144773879093121=23^{4}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.48$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{164} a^{12} + \frac{17}{82} a^{11} - \frac{9}{41} a^{10} + \frac{19}{82} a^{9} + \frac{29}{164} a^{8} + \frac{13}{82} a^{7} - \frac{31}{164} a^{6} - \frac{37}{82} a^{5} - \frac{77}{164} a^{4} - \frac{8}{41} a^{3} - \frac{29}{82} a^{2} - \frac{17}{82} a - \frac{57}{164}$, $\frac{1}{164} a^{13} + \frac{19}{82} a^{11} + \frac{8}{41} a^{10} - \frac{33}{164} a^{9} + \frac{6}{41} a^{8} - \frac{13}{164} a^{7} - \frac{1}{41} a^{6} - \frac{21}{164} a^{5} + \frac{11}{41} a^{4} + \frac{23}{82} a^{3} + \frac{13}{41} a^{2} + \frac{33}{164} a + \frac{13}{41}$, $\frac{1}{4920} a^{14} + \frac{1}{4920} a^{13} + \frac{13}{4920} a^{12} + \frac{61}{2460} a^{11} + \frac{653}{4920} a^{10} - \frac{631}{4920} a^{9} - \frac{55}{492} a^{8} - \frac{359}{1640} a^{7} - \frac{101}{615} a^{6} + \frac{145}{984} a^{5} - \frac{2249}{4920} a^{4} - \frac{43}{2460} a^{3} + \frac{703}{1640} a^{2} + \frac{1591}{4920} a + \frac{443}{984}$, $\frac{1}{233840826160969080} a^{15} + \frac{3326554111931}{116920413080484540} a^{14} + \frac{201570329436857}{116920413080484540} a^{13} + \frac{139639171175407}{46768165232193816} a^{12} - \frac{344885607713009}{46768165232193816} a^{11} - \frac{22479352730459939}{116920413080484540} a^{10} - \frac{21267999573333901}{233840826160969080} a^{9} - \frac{17419333025408629}{77946942053656360} a^{8} + \frac{9521590304943715}{46768165232193816} a^{7} - \frac{57854037170725123}{233840826160969080} a^{6} + \frac{14265429226471427}{29230103270121135} a^{5} + \frac{17307946937324093}{46768165232193816} a^{4} + \frac{17823954907456361}{77946942053656360} a^{3} - \frac{1980056943724814}{5846020654024227} a^{2} - \frac{28218578083787977}{116920413080484540} a - \frac{5246077190899603}{15589388410731272}$

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:  \( 176127.861771 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T158):

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

\(\Q(\sqrt{41}) \), 4.2.38663.1, 4.4.68921.1, 4.2.1585183.1, 8.0.61287930329.1 x2, 8.2.109252397543.1 x2, 8.4.2512805143489.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$