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Properties

Label 16.0.58231296436...7401.1
Degree $16$
Signature $[0, 8]$
Discriminant $7^{12}\cdot 29^{10}$
Root discriminant $35.30$
Ramified primes $7, 29$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_4\wr C_2$ (as 16T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![848, -2584, 1024, 4438, 429, -13571, 10883, 2352, -6198, 2585, 749, -708, 122, 42, -10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 10*x^14 + 42*x^13 + 122*x^12 - 708*x^11 + 749*x^10 + 2585*x^9 - 6198*x^8 + 2352*x^7 + 10883*x^6 - 13571*x^5 + 429*x^4 + 4438*x^3 + 1024*x^2 - 2584*x + 848)
 
gp: K = bnfinit(x^16 - 2*x^15 - 10*x^14 + 42*x^13 + 122*x^12 - 708*x^11 + 749*x^10 + 2585*x^9 - 6198*x^8 + 2352*x^7 + 10883*x^6 - 13571*x^5 + 429*x^4 + 4438*x^3 + 1024*x^2 - 2584*x + 848, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 10 x^{14} + 42 x^{13} + 122 x^{12} - 708 x^{11} + 749 x^{10} + 2585 x^{9} - 6198 x^{8} + 2352 x^{7} + 10883 x^{6} - 13571 x^{5} + 429 x^{4} + 4438 x^{3} + 1024 x^{2} - 2584 x + 848 \)

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:  \(5823129643646193092027401=7^{12}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.30$
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:  $7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{80} a^{14} - \frac{9}{80} a^{13} - \frac{9}{80} a^{12} - \frac{17}{80} a^{11} - \frac{1}{80} a^{10} - \frac{7}{80} a^{9} - \frac{1}{4} a^{8} - \frac{1}{80} a^{7} + \frac{9}{80} a^{6} + \frac{31}{80} a^{5} - \frac{3}{20} a^{4} + \frac{31}{80} a^{3} + \frac{9}{20} a^{2} - \frac{1}{5} a + \frac{3}{10}$, $\frac{1}{50855211133028842807039427360} a^{15} + \frac{46594241813527876205355291}{12713802783257210701759856840} a^{14} + \frac{2570244934415396998460358837}{25427605566514421403519713680} a^{13} + \frac{3066446789021437076375573213}{25427605566514421403519713680} a^{12} + \frac{4433199705706670142374362629}{25427605566514421403519713680} a^{11} - \frac{609193291088203725761386145}{2542760556651442140351971368} a^{10} + \frac{846737845801592385097750329}{50855211133028842807039427360} a^{9} + \frac{7424735278493329530012095219}{50855211133028842807039427360} a^{8} - \frac{496725877563210738157527571}{12713802783257210701759856840} a^{7} - \frac{4027875007358886285956061233}{12713802783257210701759856840} a^{6} + \frac{6848613541418540835506381191}{50855211133028842807039427360} a^{5} + \frac{4544469004270118809186989351}{10171042226605768561407885472} a^{4} - \frac{7660351554992404673997390281}{50855211133028842807039427360} a^{3} - \frac{5670896136006209709141049357}{12713802783257210701759856840} a^{2} - \frac{1694633350929930856506748883}{6356901391628605350879928420} a + \frac{29121207950226983519509883}{119941535691105761337357140}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 343368.044177 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-203}) \), 4.2.5887.1 x2, 4.0.1421.1 x2, \(\Q(\sqrt{-7}, \sqrt{29})\), 8.0.2869341461.1, 8.0.2413116168701.1, 8.0.1698181681.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$7$7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$29$29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$