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Properties

Label 16.0.77131955891...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 5^{8}\cdot 7841^{3}$
Root discriminant $20.20$
Ramified primes $2, 5, 7841$
Class number $1$
Class group Trivial
Galois group 16T1872

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2929, -4125, 3400, -4915, 5253, -4082, 3339, -2211, 1258, -767, 383, -162, 79, -29, 8, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 8*x^14 - 29*x^13 + 79*x^12 - 162*x^11 + 383*x^10 - 767*x^9 + 1258*x^8 - 2211*x^7 + 3339*x^6 - 4082*x^5 + 5253*x^4 - 4915*x^3 + 3400*x^2 - 4125*x + 2929)
 
gp: K = bnfinit(x^16 - 3*x^15 + 8*x^14 - 29*x^13 + 79*x^12 - 162*x^11 + 383*x^10 - 767*x^9 + 1258*x^8 - 2211*x^7 + 3339*x^6 - 4082*x^5 + 5253*x^4 - 4915*x^3 + 3400*x^2 - 4125*x + 2929, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 8 x^{14} - 29 x^{13} + 79 x^{12} - 162 x^{11} + 383 x^{10} - 767 x^{9} + 1258 x^{8} - 2211 x^{7} + 3339 x^{6} - 4082 x^{5} + 5253 x^{4} - 4915 x^{3} + 3400 x^{2} - 4125 x + 2929 \)

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:  \(771319558913600000000=2^{12}\cdot 5^{8}\cdot 7841^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.20$
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:  $2, 5, 7841$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{23} a^{14} + \frac{5}{23} a^{13} - \frac{6}{23} a^{12} - \frac{2}{23} a^{11} - \frac{4}{23} a^{10} + \frac{6}{23} a^{9} + \frac{3}{23} a^{8} - \frac{9}{23} a^{7} - \frac{11}{23} a^{6} + \frac{4}{23} a^{5} + \frac{9}{23} a^{4} + \frac{6}{23} a^{3} + \frac{8}{23} a^{2} + \frac{1}{23}$, $\frac{1}{122496359147308070209888} a^{15} - \frac{174500602268992373301}{15312044893413508776236} a^{14} - \frac{2296035784461785434805}{7656022446706754388118} a^{13} - \frac{53126500842185092136877}{122496359147308070209888} a^{12} + \frac{1581779744771298739681}{7656022446706754388118} a^{11} - \frac{12629784863947966009577}{61248179573654035104944} a^{10} + \frac{29480397086882430453593}{122496359147308070209888} a^{9} + \frac{8440397317105300133009}{30624089786827017552472} a^{8} - \frac{9733724624395632242069}{61248179573654035104944} a^{7} - \frac{35532211202283500324241}{122496359147308070209888} a^{6} - \frac{170933601720557526114}{3828011223353377194059} a^{5} - \frac{29505471198229341611801}{61248179573654035104944} a^{4} + \frac{32381794016418878670527}{122496359147308070209888} a^{3} - \frac{491625190648818887409}{2662964329289305874128} a^{2} - \frac{17369048952589296190601}{61248179573654035104944} a - \frac{2370250097056404695605}{5325928658578611748256}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 8653.46249515 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1872:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 77 conjugacy class representatives for t16n1872 are not computed
Character table for t16n1872 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.0.4900625.1

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 $16$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.12.12.10$x^{12} - 6 x^{10} + 23 x^{8} - 28 x^{6} - 9 x^{4} - 30 x^{2} - 15$$2$$6$$12$12T58$[2, 2, 2, 2]^{6}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7841Data not computed