↑

Properties

Label 16.0.35256881742...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 5^{8}\cdot 19^{8}$
Root discriminant $22.22$
Ramified primes $3, 5, 19$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^2:D_4$ (as 16T43)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, -132, 358, 484, -1715, -300, 2548, -348, -1586, 404, 478, -128, -100, 28, 16, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 16*x^14 + 28*x^13 - 100*x^12 - 128*x^11 + 478*x^10 + 404*x^9 - 1586*x^8 - 348*x^7 + 2548*x^6 - 300*x^5 - 1715*x^4 + 484*x^3 + 358*x^2 - 132*x + 49)
 
gp: K = bnfinit(x^16 - 8*x^15 + 16*x^14 + 28*x^13 - 100*x^12 - 128*x^11 + 478*x^10 + 404*x^9 - 1586*x^8 - 348*x^7 + 2548*x^6 - 300*x^5 - 1715*x^4 + 484*x^3 + 358*x^2 - 132*x + 49, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 16 x^{14} + 28 x^{13} - 100 x^{12} - 128 x^{11} + 478 x^{10} + 404 x^{9} - 1586 x^{8} - 348 x^{7} + 2548 x^{6} - 300 x^{5} - 1715 x^{4} + 484 x^{3} + 358 x^{2} - 132 x + 49 \)

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:  \(3525688174246906640625=3^{12}\cdot 5^{8}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.22$
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:  $3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{56} a^{12} - \frac{3}{28} a^{11} + \frac{3}{28} a^{10} - \frac{3}{56} a^{9} - \frac{1}{56} a^{8} + \frac{3}{14} a^{7} + \frac{1}{8} a^{6} - \frac{1}{14} a^{5} - \frac{11}{56} a^{4} + \frac{25}{56} a^{3} - \frac{23}{56} a^{2} + \frac{25}{56} a + \frac{3}{8}$, $\frac{1}{56} a^{13} - \frac{1}{28} a^{11} + \frac{5}{56} a^{10} - \frac{5}{56} a^{9} + \frac{3}{28} a^{8} - \frac{5}{56} a^{7} + \frac{5}{28} a^{6} - \frac{1}{8} a^{5} - \frac{13}{56} a^{4} - \frac{27}{56} a^{3} + \frac{27}{56} a^{2} + \frac{17}{56} a - \frac{1}{4}$, $\frac{1}{2296} a^{14} - \frac{1}{328} a^{13} - \frac{5}{1148} a^{12} + \frac{151}{2296} a^{11} - \frac{23}{1148} a^{10} - \frac{173}{2296} a^{9} + \frac{31}{2296} a^{8} + \frac{565}{2296} a^{7} + \frac{57}{328} a^{6} - \frac{44}{287} a^{5} - \frac{253}{1148} a^{4} - \frac{503}{1148} a^{3} + \frac{73}{574} a^{2} + \frac{661}{2296} a + \frac{33}{82}$, $\frac{1}{4093768} a^{15} + \frac{221}{1023442} a^{14} + \frac{5807}{4093768} a^{13} - \frac{2001}{584824} a^{12} - \frac{207691}{4093768} a^{11} + \frac{216075}{4093768} a^{10} - \frac{63216}{511721} a^{9} - \frac{50279}{511721} a^{8} + \frac{201619}{1023442} a^{7} - \frac{807767}{4093768} a^{6} - \frac{70631}{292412} a^{5} + \frac{484563}{2046884} a^{4} - \frac{180615}{1023442} a^{3} + \frac{1252869}{4093768} a^{2} - \frac{167847}{584824} a - \frac{67761}{292412}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{17}{1148} a^{14} - \frac{17}{164} a^{13} + \frac{51}{328} a^{12} + \frac{17}{41} a^{11} - \frac{141}{164} a^{10} - \frac{647}{328} a^{9} + \frac{1199}{328} a^{8} + \frac{6899}{1148} a^{7} - \frac{3761}{328} a^{6} - \frac{161}{41} a^{5} + \frac{4647}{328} a^{4} - \frac{2069}{328} a^{3} - \frac{9547}{2296} a^{2} + \frac{10051}{2296} a - \frac{145}{328} \) (order $6$)
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:  \( 23382.8111247 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_4$ (as 16T43):

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 $C_2^2:D_4$
Character table for $C_2^2:D_4$

Intermediate fields

\(\Q(\sqrt{-19}) \), \(\Q(\sqrt{57}) \), \(\Q(\sqrt{-3}) \), 4.0.16245.1, 4.0.1805.1, 4.0.12825.2 x2, \(\Q(\sqrt{-3}, \sqrt{-19})\), 4.2.243675.2 x2, 8.0.59377505625.2, 8.0.2375100225.1, 8.0.263900025.1

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 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$