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Properties

Label 16.0.47858458561...000.31
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}$
Root discriminant $169.82$
Ramified primes $2, 3, 5, 19$
Class number $11605568$ (GRH)
Class group $[2, 2, 2, 26, 55796]$ (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![271737008656, 0, 228831165184, 0, 55702323104, 0, 5823784848, 0, 295046744, 0, 7572336, 0, 96026, 0, 532, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 532*x^14 + 96026*x^12 + 7572336*x^10 + 295046744*x^8 + 5823784848*x^6 + 55702323104*x^4 + 228831165184*x^2 + 271737008656)
 
gp: K = bnfinit(x^16 + 532*x^14 + 96026*x^12 + 7572336*x^10 + 295046744*x^8 + 5823784848*x^6 + 55702323104*x^4 + 228831165184*x^2 + 271737008656, 1)
 

Normalized defining polynomial

\( x^{16} + 532 x^{14} + 96026 x^{12} + 7572336 x^{10} + 295046744 x^{8} + 5823784848 x^{6} + 55702323104 x^{4} + 228831165184 x^{2} + 271737008656 \)

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:  \(478584585616890104119296000000000000=2^{44}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $169.82$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4560=2^{4}\cdot 3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4560}(1,·)$, $\chi_{4560}(2051,·)$, $\chi_{4560}(3649,·)$, $\chi_{4560}(1673,·)$, $\chi_{4560}(1217,·)$, $\chi_{4560}(4483,·)$, $\chi_{4560}(1369,·)$, $\chi_{4560}(3419,·)$, $\chi_{4560}(2203,·)$, $\chi_{4560}(2281,·)$, $\chi_{4560}(4331,·)$, $\chi_{4560}(3953,·)$, $\chi_{4560}(1139,·)$, $\chi_{4560}(1747,·)$, $\chi_{4560}(3497,·)$, $\chi_{4560}(4027,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{19} a^{2}$, $\frac{1}{19} a^{3}$, $\frac{1}{722} a^{4}$, $\frac{1}{722} a^{5}$, $\frac{1}{13718} a^{6}$, $\frac{1}{13718} a^{7}$, $\frac{1}{521284} a^{8}$, $\frac{1}{521284} a^{9}$, $\frac{1}{9904396} a^{10}$, $\frac{1}{9904396} a^{11}$, $\frac{1}{1129101144} a^{12} + \frac{1}{29713188} a^{10} + \frac{1}{41154} a^{6} + \frac{1}{57} a^{2} - \frac{1}{3}$, $\frac{1}{1129101144} a^{13} + \frac{1}{29713188} a^{11} + \frac{1}{41154} a^{7} + \frac{1}{57} a^{3} - \frac{1}{3} a$, $\frac{1}{665040573816} a^{14} + \frac{1}{5833689244} a^{12} - \frac{13}{460554414} a^{10} + \frac{2}{12119853} a^{8} + \frac{13}{1275774} a^{6} + \frac{23}{67146} a^{4} - \frac{4}{589} a^{2} + \frac{17}{93}$, $\frac{1}{665040573816} a^{15} + \frac{1}{5833689244} a^{13} - \frac{13}{460554414} a^{11} + \frac{2}{12119853} a^{9} + \frac{13}{1275774} a^{7} + \frac{23}{67146} a^{5} - \frac{4}{589} a^{3} + \frac{17}{93} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{26}\times C_{55796}$, which has order $11605568$ (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:  \( 16694.393243512957 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.0.166348800.2, \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.6653952.2, \(\Q(\zeta_{15})^+\), 4.4.72000.1, 4.0.92416000.4, 4.0.92416000.2, 8.0.27671923261440000.256, 8.8.5184000000.1, 8.0.8540717056000000.40

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5Data not computed
$19$19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$