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Properties

Label 16.12.2738235154...2432.1
Degree $16$
Signature $[12, 2]$
Discriminant $2^{32}\cdot 17^{5}\cdot 113^{2}\cdot 593^{2}$
Root discriminant $38.89$
Ramified primes $2, 17, 113, 593$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1779

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 252, -1168, 1692, -2803, 4244, -1520, -1972, 1526, -468, 368, -32, -188, 76, 8, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 8*x^14 + 76*x^13 - 188*x^12 - 32*x^11 + 368*x^10 - 468*x^9 + 1526*x^8 - 1972*x^7 - 1520*x^6 + 4244*x^5 - 2803*x^4 + 1692*x^3 - 1168*x^2 + 252*x + 41)
 
gp: K = bnfinit(x^16 - 8*x^15 + 8*x^14 + 76*x^13 - 188*x^12 - 32*x^11 + 368*x^10 - 468*x^9 + 1526*x^8 - 1972*x^7 - 1520*x^6 + 4244*x^5 - 2803*x^4 + 1692*x^3 - 1168*x^2 + 252*x + 41, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 8 x^{14} + 76 x^{13} - 188 x^{12} - 32 x^{11} + 368 x^{10} - 468 x^{9} + 1526 x^{8} - 1972 x^{7} - 1520 x^{6} + 4244 x^{5} - 2803 x^{4} + 1692 x^{3} - 1168 x^{2} + 252 x + 41 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(27382351547454726374162432=2^{32}\cdot 17^{5}\cdot 113^{2}\cdot 593^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.89$
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:  $2, 17, 113, 593$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{73294050743978335607} a^{15} - \frac{945929622683236323}{10470578677711190801} a^{14} - \frac{14446763540162510752}{73294050743978335607} a^{13} + \frac{2005068621059457717}{10470578677711190801} a^{12} + \frac{13026224221578226611}{73294050743978335607} a^{11} - \frac{29838015559543416151}{73294050743978335607} a^{10} + \frac{1451307828739484052}{73294050743978335607} a^{9} - \frac{21512060262196846067}{73294050743978335607} a^{8} - \frac{11538192124880583188}{73294050743978335607} a^{7} + \frac{26143793719775529207}{73294050743978335607} a^{6} + \frac{35370727322977118824}{73294050743978335607} a^{5} - \frac{34654427198191051414}{73294050743978335607} a^{4} + \frac{3537260991077260141}{73294050743978335607} a^{3} - \frac{2652841270137568942}{10470578677711190801} a^{2} - \frac{6572471362464417619}{73294050743978335607} a + \frac{21158926627621731271}{73294050743978335607}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 15805300.7777 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1779:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 136 conjugacy class representatives for t16n1779 are not computed
Character table for t16n1779 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.4352.1, 8.8.1269144027136.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.3.4$x^{4} + 459$$4$$1$$3$$C_4$$[\ ]_{4}$
17.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$113$113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
113.8.0.1$x^{8} - 2 x + 5$$1$$8$$0$$C_8$$[\ ]^{8}$
593Data not computed