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Properties

Label 16.0.45249614645...000.21
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 17^{14}$
Root discriminant $195.42$
Ramified primes $2, 3, 5, 17$
Class number $13508608$ (GRH)
Class group $[2, 2, 2, 4, 4, 8, 13192]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![303380640000, 0, 252817200000, 0, 62361576000, 0, 6179976000, 0, 294127200, 0, 7160400, 0, 88740, 0, 510, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 510*x^14 + 88740*x^12 + 7160400*x^10 + 294127200*x^8 + 6179976000*x^6 + 62361576000*x^4 + 252817200000*x^2 + 303380640000)
 
gp: K = bnfinit(x^16 + 510*x^14 + 88740*x^12 + 7160400*x^10 + 294127200*x^8 + 6179976000*x^6 + 62361576000*x^4 + 252817200000*x^2 + 303380640000, 1)
 

Normalized defining polynomial

\( x^{16} + 510 x^{14} + 88740 x^{12} + 7160400 x^{10} + 294127200 x^{8} + 6179976000 x^{6} + 62361576000 x^{4} + 252817200000 x^{2} + 303380640000 \)

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:  \(4524961464550316012212224000000000000=2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $195.42$
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:  $2, 3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2040=2^{3}\cdot 3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{2040}(1,·)$, $\chi_{2040}(1283,·)$, $\chi_{2040}(769,·)$, $\chi_{2040}(1801,·)$, $\chi_{2040}(587,·)$, $\chi_{2040}(563,·)$, $\chi_{2040}(83,·)$, $\chi_{2040}(409,·)$, $\chi_{2040}(1307,·)$, $\chi_{2040}(1441,·)$, $\chi_{2040}(1787,·)$, $\chi_{2040}(169,·)$, $\chi_{2040}(467,·)$, $\chi_{2040}(361,·)$, $\chi_{2040}(1849,·)$, $\chi_{2040}(1403,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{6} a^{2}$, $\frac{1}{6} a^{3}$, $\frac{1}{180} a^{4}$, $\frac{1}{180} a^{5}$, $\frac{1}{4320} a^{6} - \frac{1}{4}$, $\frac{1}{4320} a^{7} - \frac{1}{4} a$, $\frac{1}{2203200} a^{8} - \frac{1}{24} a^{2}$, $\frac{1}{2203200} a^{9} - \frac{1}{24} a^{3}$, $\frac{1}{26438400} a^{10} - \frac{1}{4406400} a^{8} - \frac{1}{8640} a^{6} - \frac{1}{1440} a^{4} - \frac{1}{16} a^{2} + \frac{1}{8}$, $\frac{1}{26438400} a^{11} - \frac{1}{4406400} a^{9} - \frac{1}{8640} a^{7} - \frac{1}{1440} a^{5} - \frac{1}{16} a^{3} + \frac{1}{8} a$, $\frac{1}{41243904000} a^{12} - \frac{1}{85924800} a^{10} + \frac{1}{14320800} a^{8} - \frac{7}{224640} a^{6} + \frac{1}{1560} a^{4} + \frac{1}{52} a^{2} + \frac{81}{416}$, $\frac{1}{41243904000} a^{13} - \frac{1}{85924800} a^{11} + \frac{1}{14320800} a^{9} - \frac{7}{224640} a^{7} + \frac{1}{1560} a^{5} + \frac{1}{52} a^{3} + \frac{81}{416} a$, $\frac{1}{247463424000} a^{14} - \frac{1}{85924800} a^{10} - \frac{7}{114566400} a^{8} - \frac{1}{14040} a^{6} - \frac{1}{936} a^{4} - \frac{45}{832} a^{2} + \frac{1}{13}$, $\frac{1}{247463424000} a^{15} - \frac{1}{85924800} a^{11} - \frac{7}{114566400} a^{9} - \frac{1}{14040} a^{7} - \frac{1}{936} a^{5} - \frac{45}{832} a^{3} + \frac{1}{13} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{8}\times C_{13192}$, which has order $13508608$ (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:  \( 55081.08216847024 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

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 $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}, \sqrt{17})\), 4.4.122825.1, 4.4.4913.1, 8.8.15085980625.1, 8.0.2127195680832000000.1, 8.0.2127195680832000000.2

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
5Data not computed
17Data not computed