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Properties

Label 16.0.32790240335...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 7^{8}\cdot 13^{12}$
Root discriminant $60.57$
Ramified primes $5, 7, 13$
Class number $160$ (GRH)
Class group $[4, 40]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![250721, 40056, 772584, -416788, -77783, 192150, -37967, -39196, 13769, 4356, -2816, -376, 458, -2, -31, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 31*x^14 - 2*x^13 + 458*x^12 - 376*x^11 - 2816*x^10 + 4356*x^9 + 13769*x^8 - 39196*x^7 - 37967*x^6 + 192150*x^5 - 77783*x^4 - 416788*x^3 + 772584*x^2 + 40056*x + 250721)
 
gp: K = bnfinit(x^16 - 31*x^14 - 2*x^13 + 458*x^12 - 376*x^11 - 2816*x^10 + 4356*x^9 + 13769*x^8 - 39196*x^7 - 37967*x^6 + 192150*x^5 - 77783*x^4 - 416788*x^3 + 772584*x^2 + 40056*x + 250721, 1)
 

Normalized defining polynomial

\( x^{16} - 31 x^{14} - 2 x^{13} + 458 x^{12} - 376 x^{11} - 2816 x^{10} + 4356 x^{9} + 13769 x^{8} - 39196 x^{7} - 37967 x^{6} + 192150 x^{5} - 77783 x^{4} - 416788 x^{3} + 772584 x^{2} + 40056 x + 250721 \)

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:  \(32790240335000876777587890625=5^{12}\cdot 7^{8}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.57$
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:  $5, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(455=5\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{455}(64,·)$, $\chi_{455}(1,·)$, $\chi_{455}(118,·)$, $\chi_{455}(398,·)$, $\chi_{455}(272,·)$, $\chi_{455}(274,·)$, $\chi_{455}(83,·)$, $\chi_{455}(281,·)$, $\chi_{455}(27,·)$, $\chi_{455}(99,·)$, $\chi_{455}(421,·)$, $\chi_{455}(363,·)$, $\chi_{455}(239,·)$, $\chi_{455}(307,·)$, $\chi_{455}(246,·)$, $\chi_{455}(447,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{10} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{1}{12} a^{11} + \frac{1}{12} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{5}{12}$, $\frac{1}{48} a^{14} - \frac{1}{24} a^{13} + \frac{1}{6} a^{11} + \frac{1}{24} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{13}{48} a^{6} - \frac{7}{24} a^{5} + \frac{7}{24} a^{4} + \frac{1}{3} a^{3} - \frac{7}{16} a^{2} + \frac{7}{24} a + \frac{13}{48}$, $\frac{1}{8484937210756062960964877417978707392} a^{15} + \frac{36787977974391783677339990091494569}{8484937210756062960964877417978707392} a^{14} + \frac{144414011817764457756538809430927081}{4242468605378031480482438708989353696} a^{13} - \frac{4689309368096254229115433142157599}{88384762612042322510050806437278202} a^{12} + \frac{331187221361578696400683986107378759}{1414156201792677160160812902996451232} a^{11} - \frac{879091858448939993635644741519231311}{4242468605378031480482438708989353696} a^{10} + \frac{14723761160659734149883221293772867}{1414156201792677160160812902996451232} a^{9} + \frac{10746902675511635640556185042118081}{1414156201792677160160812902996451232} a^{8} - \frac{1581216765888500372286145865883555457}{8484937210756062960964877417978707392} a^{7} - \frac{1779789899114705441328803945560278917}{8484937210756062960964877417978707392} a^{6} + \frac{48660543869046882186025524372998243}{707078100896338580080406451498225616} a^{5} + \frac{4060180353851426047571934779224321}{15771258756052161637481184791781984} a^{4} - \frac{824579379432473661793844099394621485}{8484937210756062960964877417978707392} a^{3} + \frac{1442386956240614872586001958409082839}{8484937210756062960964877417978707392} a^{2} - \frac{4058074838147083421870210526162692585}{8484937210756062960964877417978707392} a + \frac{3683584027979322908586131266284882247}{8484937210756062960964877417978707392}$

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

Class group and class number

$C_{4}\times C_{40}$, which has order $160$ (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:  \( 162215.76213779865 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), 4.0.13456625.1, \(\Q(\sqrt{5}, \sqrt{13})\), 4.0.13456625.2, 4.4.1035125.1, 4.4.6125.1, 4.0.2197.1, 4.0.54925.1, 8.0.181080756390625.7, 8.8.1071483765625.1, 8.0.3016755625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
13Data not computed