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Properties

Label 16.0.10236206731...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 109^{6}$
Root discriminant $23.75$
Ramified primes $5, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\wr C_2$ (as 16T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![751, 409, -1846, -2006, 946, 2427, 1121, -623, -1098, -317, 281, 108, -44, 11, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^14 + 11*x^13 - 44*x^12 + 108*x^11 + 281*x^10 - 317*x^9 - 1098*x^8 - 623*x^7 + 1121*x^6 + 2427*x^5 + 946*x^4 - 2006*x^3 - 1846*x^2 + 409*x + 751)
 
gp: K = bnfinit(x^16 - 4*x^15 + 14*x^14 + 11*x^13 - 44*x^12 + 108*x^11 + 281*x^10 - 317*x^9 - 1098*x^8 - 623*x^7 + 1121*x^6 + 2427*x^5 + 946*x^4 - 2006*x^3 - 1846*x^2 + 409*x + 751, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 14 x^{14} + 11 x^{13} - 44 x^{12} + 108 x^{11} + 281 x^{10} - 317 x^{9} - 1098 x^{8} - 623 x^{7} + 1121 x^{6} + 2427 x^{5} + 946 x^{4} - 2006 x^{3} - 1846 x^{2} + 409 x + 751 \)

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:  \(10236206731207275390625=5^{14}\cdot 109^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.75$
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:  $5, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2882} a^{14} + \frac{233}{1441} a^{13} - \frac{3}{262} a^{12} - \frac{107}{2882} a^{11} - \frac{185}{2882} a^{10} - \frac{480}{1441} a^{9} - \frac{28}{131} a^{8} - \frac{590}{1441} a^{7} - \frac{44}{131} a^{6} + \frac{725}{2882} a^{5} - \frac{175}{1441} a^{4} + \frac{1323}{2882} a^{3} + \frac{483}{1441} a^{2} + \frac{394}{1441} a - \frac{1049}{2882}$, $\frac{1}{2698644120914344922} a^{15} + \frac{104292533972590}{1349322060457172461} a^{14} - \frac{295199125734802775}{1349322060457172461} a^{13} + \frac{245053041086819881}{2698644120914344922} a^{12} - \frac{460033478636852891}{2698644120914344922} a^{11} + \frac{100127039247354672}{1349322060457172461} a^{10} - \frac{440452433990594561}{2698644120914344922} a^{9} + \frac{526593218433339109}{2698644120914344922} a^{8} + \frac{1190353997909553337}{2698644120914344922} a^{7} + \frac{368410721021156753}{1349322060457172461} a^{6} - \frac{128887977571908086}{1349322060457172461} a^{5} - \frac{183436820184752095}{1349322060457172461} a^{4} - \frac{424856035771553460}{1349322060457172461} a^{3} - \frac{1249353821749037501}{2698644120914344922} a^{2} - \frac{51488302697661286}{122665641859742951} a + \frac{221100369113563205}{1349322060457172461}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( -\frac{3495507769165}{122665641859742951} a^{15} + \frac{378068766412703}{245331283719485902} a^{14} - \frac{940184871671385}{122665641859742951} a^{13} + \frac{3407042162731091}{122665641859742951} a^{12} - \frac{2415685987819938}{122665641859742951} a^{11} - \frac{5878486676530361}{245331283719485902} a^{10} + \frac{25598424409807715}{245331283719485902} a^{9} + \frac{45228881768429519}{245331283719485902} a^{8} - \frac{127503576633713519}{245331283719485902} a^{7} - \frac{193589205154259659}{122665641859742951} a^{6} - \frac{270475852771968033}{245331283719485902} a^{5} + \frac{209064444213541234}{122665641859742951} a^{4} + \frac{935540082424907723}{245331283719485902} a^{3} + \frac{675324379191014099}{245331283719485902} a^{2} - \frac{144338665822787626}{122665641859742951} a - \frac{254460024016398349}{122665641859742951} \) (order $10$)
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:  \( 118928.142177 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T28):

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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.13625.1, 4.0.2725.1, 8.0.185640625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
$109$109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$
109.4.2.2$x^{4} - 109 x^{2} + 71286$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
109.8.4.1$x^{8} + 712860 x^{4} - 1295029 x^{2} + 127042344900$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$