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Properties

Label 16.0.86844229865...336.63
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 7^{8}\cdot 13^{8}$
Root discriminant $132.18$
Ramified primes $2, 3, 7, 13$
Class number $1916928$ (GRH)
Class group $[2, 4, 4, 48, 1248]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![148857727873, -40816151264, 45933051828, -10616609960, 6284020592, -1223811416, 498037052, -80960624, 24998649, -3318856, 812720, -84344, 16666, -1232, 196, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 196*x^14 - 1232*x^13 + 16666*x^12 - 84344*x^11 + 812720*x^10 - 3318856*x^9 + 24998649*x^8 - 80960624*x^7 + 498037052*x^6 - 1223811416*x^5 + 6284020592*x^4 - 10616609960*x^3 + 45933051828*x^2 - 40816151264*x + 148857727873)
 
gp: K = bnfinit(x^16 - 8*x^15 + 196*x^14 - 1232*x^13 + 16666*x^12 - 84344*x^11 + 812720*x^10 - 3318856*x^9 + 24998649*x^8 - 80960624*x^7 + 498037052*x^6 - 1223811416*x^5 + 6284020592*x^4 - 10616609960*x^3 + 45933051828*x^2 - 40816151264*x + 148857727873, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 196 x^{14} - 1232 x^{13} + 16666 x^{12} - 84344 x^{11} + 812720 x^{10} - 3318856 x^{9} + 24998649 x^{8} - 80960624 x^{7} + 498037052 x^{6} - 1223811416 x^{5} + 6284020592 x^{4} - 10616609960 x^{3} + 45933051828 x^{2} - 40816151264 x + 148857727873 \)

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:  \(8684422986556880818624724610318336=2^{48}\cdot 3^{8}\cdot 7^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $132.18$
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:  $2, 3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4368=2^{4}\cdot 3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{4368}(1,·)$, $\chi_{4368}(1091,·)$, $\chi_{4368}(1093,·)$, $\chi_{4368}(3457,·)$, $\chi_{4368}(2185,·)$, $\chi_{4368}(3275,·)$, $\chi_{4368}(3277,·)$, $\chi_{4368}(911,·)$, $\chi_{4368}(2003,·)$, $\chi_{4368}(3095,·)$, $\chi_{4368}(4367,·)$, $\chi_{4368}(4187,·)$, $\chi_{4368}(2183,·)$, $\chi_{4368}(181,·)$, $\chi_{4368}(1273,·)$, $\chi_{4368}(2365,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{534789486764709689955289} a^{14} - \frac{7}{534789486764709689955289} a^{13} + \frac{107380807712300708602734}{534789486764709689955289} a^{12} - \frac{109495359509094561661024}{534789486764709689955289} a^{11} + \frac{205447921343364137384671}{534789486764709689955289} a^{10} + \frac{65599436577331076628413}{534789486764709689955289} a^{9} - \frac{123738721702557462200862}{534789486764709689955289} a^{8} - \frac{33511713724377409327430}{534789486764709689955289} a^{7} - \frac{12808662620081044249584}{534789486764709689955289} a^{6} + \frac{191751959855678602625775}{534789486764709689955289} a^{5} - \frac{149810520299934824176245}{534789486764709689955289} a^{4} + \frac{98829902315132240627372}{534789486764709689955289} a^{3} - \frac{239571563030597476558247}{534789486764709689955289} a^{2} - \frac{73486917163987695567}{534789486764709689955289} a - \frac{265121096633876616013643}{534789486764709689955289}$, $\frac{1}{51036347431095426604659283334377} a^{15} + \frac{47716289}{51036347431095426604659283334377} a^{14} - \frac{3411829538280190108708159872556}{51036347431095426604659283334377} a^{13} - \frac{23735640973064563589210258999509}{51036347431095426604659283334377} a^{12} + \frac{1125600058756207524916587523791}{51036347431095426604659283334377} a^{11} - \frac{1128496085218381027428091994165}{51036347431095426604659283334377} a^{10} - \frac{6136795892769844374479812767529}{51036347431095426604659283334377} a^{9} + \frac{11617287089000381673669733722176}{51036347431095426604659283334377} a^{8} + \frac{537145696848712415973848202966}{51036347431095426604659283334377} a^{7} + \frac{13456411316626324127645965460924}{51036347431095426604659283334377} a^{6} - \frac{13615084028143012250125871047260}{51036347431095426604659283334377} a^{5} - \frac{17295705182331289371121398188702}{51036347431095426604659283334377} a^{4} + \frac{11781988839202518140539813487980}{51036347431095426604659283334377} a^{3} + \frac{790208619213008262371275684704}{51036347431095426604659283334377} a^{2} - \frac{8572380977778993724805729349263}{51036347431095426604659283334377} a + \frac{4643048673582352596257387776927}{51036347431095426604659283334377}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{48}\times C_{1248}$, which has order $1916928$ (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:  \( 11964.310642723332 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-546}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-182}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-273}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-91}) \), \(\Q(\sqrt{3}, \sqrt{-182})\), \(\Q(\sqrt{2}, \sqrt{-273})\), \(\Q(\sqrt{6}, \sqrt{-91})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-91})\), \(\Q(\sqrt{2}, \sqrt{-91})\), \(\Q(\sqrt{6}, \sqrt{-182})\), \(\Q(\zeta_{16})^+\), 4.0.152635392.5, 4.4.18432.1, 4.0.16959488.5, 8.0.364024420171776.258, 8.0.93190251563974656.25, 8.0.93190251563974656.17, \(\Q(\zeta_{48})^+\), 8.0.93190251563974656.23, 8.0.287624233222144.127, 8.0.23297562890993664.177

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$