Properties

Label 16.0.23612624896...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{12}\cdot 7^{8}$
Root discriminant $25.02$
Ramified primes $2, 5, 7$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -480, 4376, -6272, 372, -264, 8262, -11960, 10467, -7148, 3898, -1752, 656, -196, 48, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 48*x^14 - 196*x^13 + 656*x^12 - 1752*x^11 + 3898*x^10 - 7148*x^9 + 10467*x^8 - 11960*x^7 + 8262*x^6 - 264*x^5 + 372*x^4 - 6272*x^3 + 4376*x^2 - 480*x + 16)
 
gp: K = bnfinit(x^16 - 8*x^15 + 48*x^14 - 196*x^13 + 656*x^12 - 1752*x^11 + 3898*x^10 - 7148*x^9 + 10467*x^8 - 11960*x^7 + 8262*x^6 - 264*x^5 + 372*x^4 - 6272*x^3 + 4376*x^2 - 480*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 656 x^{12} - 1752 x^{11} + 3898 x^{10} - 7148 x^{9} + 10467 x^{8} - 11960 x^{7} + 8262 x^{6} - 264 x^{5} + 372 x^{4} - 6272 x^{3} + 4376 x^{2} - 480 x + 16 \)

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:  \(23612624896000000000000=2^{24}\cdot 5^{12}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.02$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(280=2^{3}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{280}(1,·)$, $\chi_{280}(69,·)$, $\chi_{280}(13,·)$, $\chi_{280}(141,·)$, $\chi_{280}(209,·)$, $\chi_{280}(153,·)$, $\chi_{280}(29,·)$, $\chi_{280}(197,·)$, $\chi_{280}(97,·)$, $\chi_{280}(41,·)$, $\chi_{280}(237,·)$, $\chi_{280}(113,·)$, $\chi_{280}(181,·)$, $\chi_{280}(169,·)$, $\chi_{280}(57,·)$, $\chi_{280}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7}$, $\frac{1}{40} a^{12} + \frac{1}{10} a^{11} - \frac{3}{40} a^{10} - \frac{1}{40} a^{8} + \frac{1}{5} a^{7} - \frac{9}{40} a^{6} + \frac{1}{10} a^{5} - \frac{1}{10} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{40} a^{13} + \frac{1}{40} a^{11} + \frac{1}{20} a^{10} - \frac{1}{40} a^{9} + \frac{1}{20} a^{8} - \frac{1}{40} a^{7} - \frac{1}{4} a^{6} + \frac{1}{20} a^{4} - \frac{1}{5} a^{3} + \frac{1}{10} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{847507160} a^{14} - \frac{7}{847507160} a^{13} + \frac{8689041}{847507160} a^{12} - \frac{10426831}{169501432} a^{11} + \frac{40085}{936472} a^{10} + \frac{84634839}{847507160} a^{9} - \frac{456891}{8921128} a^{8} - \frac{60158713}{847507160} a^{7} - \frac{214483}{1389356} a^{6} - \frac{15079731}{105938395} a^{5} - \frac{35775893}{211876790} a^{4} + \frac{19241563}{42375358} a^{3} - \frac{104233}{585295} a^{2} + \frac{23346361}{105938395} a + \frac{7502711}{105938395}$, $\frac{1}{2801011163800} a^{15} + \frac{329}{560202232760} a^{14} - \frac{2336899791}{1400505581900} a^{13} + \frac{9725622913}{2801011163800} a^{12} - \frac{8407651763}{280101116380} a^{11} + \frac{274560552493}{2801011163800} a^{10} + \frac{92991838641}{1400505581900} a^{9} - \frac{214696714787}{2801011163800} a^{8} - \frac{343319650759}{2801011163800} a^{7} - \frac{45265879711}{1400505581900} a^{6} - \frac{65514602683}{350126395475} a^{5} - \frac{44892454613}{1400505581900} a^{4} - \frac{83081896592}{350126395475} a^{3} - \frac{29662287261}{70025279095} a^{2} - \frac{127620462993}{350126395475} a - \frac{173188401789}{350126395475}$

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

Class group and class number

$C_{4}$, 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:  \( \frac{3162903}{423753580} a^{14} - \frac{22140321}{423753580} a^{13} + \frac{259143999}{847507160} a^{12} - \frac{122401956}{105938395} a^{11} + \frac{17468601}{4682360} a^{10} - \frac{3944147883}{423753580} a^{9} + \frac{877017393}{44605640} a^{8} - \frac{7071434721}{211876790} a^{7} + \frac{612123373}{13893560} a^{6} - \frac{9319400199}{211876790} a^{5} + \frac{6724200103}{423753580} a^{4} + \frac{1697853116}{105938395} a^{3} + \frac{20065621}{1170590} a^{2} - \frac{3064218226}{105938395} a + \frac{272112387}{105938395} \) (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:  \( 34736.104354 \) (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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{2}, \sqrt{-35})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{10}, \sqrt{-14})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{5}, \sqrt{-14})\), \(\Q(\sqrt{-7}, \sqrt{10})\), \(\Q(\zeta_{5})\), 4.4.6125.1, 4.0.8000.2, 4.4.392000.1, 8.0.6146560000.2, 8.0.37515625.1, 8.0.153664000000.2, 8.0.64000000.2, 8.8.153664000000.1, 8.0.153664000000.5, 8.0.153664000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$