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Properties

Label 16.0.12213777651...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 17^{2}\cdot 19^{4}\cdot 103^{8}$
Root discriminant $135.03$
Ramified primes $2, 5, 17, 19, 103$
Class number $2403776$ (GRH)
Class group $[2, 2, 2, 300472]$ (GRH)
Galois group 16T1665

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32527204609, 0, 51865194328, 0, 19807098481, 0, 2344543474, 0, 125045630, 0, 3419732, 0, 49410, 0, 356, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 356*x^14 + 49410*x^12 + 3419732*x^10 + 125045630*x^8 + 2344543474*x^6 + 19807098481*x^4 + 51865194328*x^2 + 32527204609)
 
gp: K = bnfinit(x^16 + 356*x^14 + 49410*x^12 + 3419732*x^10 + 125045630*x^8 + 2344543474*x^6 + 19807098481*x^4 + 51865194328*x^2 + 32527204609, 1)
 

Normalized defining polynomial

\( x^{16} + 356 x^{14} + 49410 x^{12} + 3419732 x^{10} + 125045630 x^{8} + 2344543474 x^{6} + 19807098481 x^{4} + 51865194328 x^{2} + 32527204609 \)

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:  \(12213777651561724066555110400000000=2^{16}\cdot 5^{8}\cdot 17^{2}\cdot 19^{4}\cdot 103^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $135.03$
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:  $2, 5, 17, 19, 103$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{618} a^{10} - \frac{28}{309} a^{8} - \frac{5}{103} a^{6} + \frac{29}{618} a^{4} - \frac{181}{618} a^{2} - \frac{1}{3}$, $\frac{1}{618} a^{11} - \frac{28}{309} a^{9} - \frac{5}{103} a^{7} + \frac{29}{618} a^{5} - \frac{181}{618} a^{3} - \frac{1}{3} a$, $\frac{1}{63654} a^{12} + \frac{47}{63654} a^{10} + \frac{13669}{63654} a^{8} + \frac{23513}{63654} a^{6} + \frac{4802}{31827} a^{4} + \frac{34}{103} a^{2} + \frac{1}{6}$, $\frac{1}{63654} a^{13} + \frac{47}{63654} a^{11} + \frac{13669}{63654} a^{9} + \frac{23513}{63654} a^{7} + \frac{4802}{31827} a^{5} + \frac{34}{103} a^{3} + \frac{1}{6} a$, $\frac{1}{25418385721569802135263354057894} a^{14} - \frac{64857627514687115771057815}{12709192860784901067631677028947} a^{12} + \frac{15292381378696333105554411511}{25418385721569802135263354057894} a^{10} - \frac{704494215716584149537534754187}{8472795240523267378421118019298} a^{8} - \frac{4129506215303457718972216046365}{25418385721569802135263354057894} a^{6} - \frac{222978660866494068901868783}{4838832233308547903153122798} a^{4} + \frac{42868127682182436121277651}{126101402094397518171082914} a^{2} - \frac{163165372374314009602847}{684159520215177854154933}$, $\frac{1}{25418385721569802135263354057894} a^{15} - \frac{64857627514687115771057815}{12709192860784901067631677028947} a^{13} + \frac{15292381378696333105554411511}{25418385721569802135263354057894} a^{11} - \frac{704494215716584149537534754187}{8472795240523267378421118019298} a^{9} - \frac{4129506215303457718972216046365}{25418385721569802135263354057894} a^{7} - \frac{222978660866494068901868783}{4838832233308547903153122798} a^{5} + \frac{42868127682182436121277651}{126101402094397518171082914} a^{3} - \frac{163165372374314009602847}{684159520215177854154933} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{300472}$, which has order $2403776$ (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:  \( 10491.0986114 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1665:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6144
The 78 conjugacy class representatives for t16n1665 are not computed
Character table for t16n1665 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.1957.1, 8.8.2393655625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.8.8.7$x^{8} + 2 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.8.8.7$x^{8} + 2 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
5Data not computed
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
$103$103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.8.4.1$x^{8} + 106090 x^{4} - 1092727 x^{2} + 2813772025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$