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Properties

Label 16.0.93219557956...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 17^{12}$
Root discriminant $55.99$
Ramified primes $2, 5, 17$
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![3481, 0, -1878, 0, 52041, 0, -20064, 0, -1666, 0, 1218, 0, 264, 0, 24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 24*x^14 + 264*x^12 + 1218*x^10 - 1666*x^8 - 20064*x^6 + 52041*x^4 - 1878*x^2 + 3481)
 
gp: K = bnfinit(x^16 + 24*x^14 + 264*x^12 + 1218*x^10 - 1666*x^8 - 20064*x^6 + 52041*x^4 - 1878*x^2 + 3481, 1)
 

Normalized defining polynomial

\( x^{16} + 24 x^{14} + 264 x^{12} + 1218 x^{10} - 1666 x^{8} - 20064 x^{6} + 52041 x^{4} - 1878 x^{2} + 3481 \)

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:  \(9321955795676176000000000000=2^{16}\cdot 5^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.99$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(340=2^{2}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{340}(1,·)$, $\chi_{340}(259,·)$, $\chi_{340}(69,·)$, $\chi_{340}(203,·)$, $\chi_{340}(13,·)$, $\chi_{340}(67,·)$, $\chi_{340}(217,·)$, $\chi_{340}(157,·)$, $\chi_{340}(293,·)$, $\chi_{340}(101,·)$, $\chi_{340}(103,·)$, $\chi_{340}(169,·)$, $\chi_{340}(191,·)$, $\chi_{340}(307,·)$, $\chi_{340}(251,·)$, $\chi_{340}(319,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{12} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} + \frac{1}{12}$, $\frac{1}{12} a^{9} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} + \frac{1}{12} a$, $\frac{1}{228} a^{10} - \frac{3}{76} a^{8} + \frac{5}{38} a^{6} - \frac{31}{76} a^{4} + \frac{1}{57} a^{2} - \frac{3}{76}$, $\frac{1}{228} a^{11} - \frac{3}{76} a^{9} + \frac{5}{38} a^{7} - \frac{31}{76} a^{5} + \frac{1}{57} a^{3} - \frac{3}{76} a$, $\frac{1}{479484} a^{12} - \frac{11}{479484} a^{10} + \frac{4603}{239742} a^{8} - \frac{6401}{53276} a^{6} + \frac{2531}{12618} a^{4} + \frac{96085}{479484} a^{2} - \frac{39430}{119871}$, $\frac{1}{28289556} a^{13} - \frac{4217}{28289556} a^{11} - \frac{16427}{14144778} a^{9} + \frac{405787}{3143284} a^{7} - \frac{2874941}{7072389} a^{5} + \frac{5593327}{28289556} a^{3} - \frac{1008913}{7072389} a$, $\frac{1}{537501564} a^{14} + \frac{5}{29861198} a^{12} + \frac{664231}{537501564} a^{10} + \frac{620111}{268750782} a^{8} + \frac{187055473}{537501564} a^{6} + \frac{27953729}{59722396} a^{4} - \frac{65896831}{268750782} a^{2} + \frac{1678559}{9110196}$, $\frac{1}{537501564} a^{15} - \frac{5}{537501564} a^{13} + \frac{532423}{268750782} a^{11} + \frac{363446}{44791797} a^{9} - \frac{39973103}{134375391} a^{7} - \frac{268443551}{537501564} a^{5} - \frac{125658163}{537501564} a^{3} - \frac{18359881}{179167188} a$

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:  \( 229864.8182635129 \) (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

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}) \), 4.0.614125.2, \(\Q(\sqrt{5}, \sqrt{17})\), 4.0.614125.1, 4.0.78608.1, 4.0.1965200.1, \(\Q(\zeta_{20})^+\), 4.4.578000.1, 8.0.377149515625.1, 8.0.3862011040000.4, 8.8.334084000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
5Data not computed
17Data not computed