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Properties

Label 16.0.10530570899...6721.2
Degree $16$
Signature $[0, 8]$
Discriminant $11^{8}\cdot 53^{12}$
Root discriminant $65.15$
Ramified primes $11, 53$
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![293423, -340957, 572377, -51675, 37741, 78685, -7623, 4376, 3661, -1558, 1237, -590, 116, -66, 17, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 17*x^14 - 66*x^13 + 116*x^12 - 590*x^11 + 1237*x^10 - 1558*x^9 + 3661*x^8 + 4376*x^7 - 7623*x^6 + 78685*x^5 + 37741*x^4 - 51675*x^3 + 572377*x^2 - 340957*x + 293423)
 
gp: K = bnfinit(x^16 - 2*x^15 + 17*x^14 - 66*x^13 + 116*x^12 - 590*x^11 + 1237*x^10 - 1558*x^9 + 3661*x^8 + 4376*x^7 - 7623*x^6 + 78685*x^5 + 37741*x^4 - 51675*x^3 + 572377*x^2 - 340957*x + 293423, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 17 x^{14} - 66 x^{13} + 116 x^{12} - 590 x^{11} + 1237 x^{10} - 1558 x^{9} + 3661 x^{8} + 4376 x^{7} - 7623 x^{6} + 78685 x^{5} + 37741 x^{4} - 51675 x^{3} + 572377 x^{2} - 340957 x + 293423 \)

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:  \(105305708997757955232277716721=11^{8}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.15$
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:  $11, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{112732830015351287337851807325000041431491523} a^{15} - \frac{12400064246420239887578398900824577481174254}{112732830015351287337851807325000041431491523} a^{14} - \frac{1550111302012598372323826197285378844252284}{112732830015351287337851807325000041431491523} a^{13} + \frac{39322722175205206406049171874306904594385176}{112732830015351287337851807325000041431491523} a^{12} - \frac{32528782430940846129924099172589887450497722}{112732830015351287337851807325000041431491523} a^{11} - \frac{10583238825780900330001170071302496046370087}{112732830015351287337851807325000041431491523} a^{10} - \frac{24525646566877004818199894245992105530248544}{112732830015351287337851807325000041431491523} a^{9} + \frac{24411445137149923558273837993135069939508334}{112732830015351287337851807325000041431491523} a^{8} + \frac{31501161923108496628845125869737048016794755}{112732830015351287337851807325000041431491523} a^{7} + \frac{2014925939232256087177992986030225217811276}{112732830015351287337851807325000041431491523} a^{6} - \frac{20811824195489577647186113030427308401301221}{112732830015351287337851807325000041431491523} a^{5} + \frac{56252359875653087656810314042746973108287864}{112732830015351287337851807325000041431491523} a^{4} + \frac{53141234366259875816944584620596832759367477}{112732830015351287337851807325000041431491523} a^{3} + \frac{28090612355786116861611185518516831102025905}{112732830015351287337851807325000041431491523} a^{2} - \frac{35427089380847686046447109395407629164038696}{112732830015351287337851807325000041431491523} a - \frac{50185710614787331023925615633428548644528150}{112732830015351287337851807325000041431491523}$

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:  \( -1 \) (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:  \( 30970947.2943 \) (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

\(\Q(\sqrt{53}) \), 4.2.30899.1, 4.0.148877.1, 4.2.1637647.1, 8.4.6122800213013.1, 8.0.6122800213013.1, 8.0.2681887696609.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$53$53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$