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Properties

Label 16.8.49023798342...0625.2
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 439^{4}\cdot 2411^{4}$
Root discriminant $71.72$
Ramified primes $5, 439, 2411$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1869

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21906751, 48594476, 41038390, 13271587, -3651274, -5295348, -2162497, -328690, 113753, 82573, 24032, 4169, -486, -49, -10, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 10*x^14 - 49*x^13 - 486*x^12 + 4169*x^11 + 24032*x^10 + 82573*x^9 + 113753*x^8 - 328690*x^7 - 2162497*x^6 - 5295348*x^5 - 3651274*x^4 + 13271587*x^3 + 41038390*x^2 + 48594476*x + 21906751)
 
gp: K = bnfinit(x^16 - 7*x^15 - 10*x^14 - 49*x^13 - 486*x^12 + 4169*x^11 + 24032*x^10 + 82573*x^9 + 113753*x^8 - 328690*x^7 - 2162497*x^6 - 5295348*x^5 - 3651274*x^4 + 13271587*x^3 + 41038390*x^2 + 48594476*x + 21906751, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 10 x^{14} - 49 x^{13} - 486 x^{12} + 4169 x^{11} + 24032 x^{10} + 82573 x^{9} + 113753 x^{8} - 328690 x^{7} - 2162497 x^{6} - 5295348 x^{5} - 3651274 x^{4} + 13271587 x^{3} + 41038390 x^{2} + 48594476 x + 21906751 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(490237983424834767085031640625=5^{8}\cdot 439^{4}\cdot 2411^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.72$
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:  $5, 439, 2411$
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}{30046336950257786029355987498554718011503310551884202013} a^{15} + \frac{4724068296709875013148042421763446004276775368960991457}{30046336950257786029355987498554718011503310551884202013} a^{14} + \frac{2594179003608861734677431555631905073248740691923030751}{30046336950257786029355987498554718011503310551884202013} a^{13} - \frac{687826489688023901785101423172600412284039896935432656}{30046336950257786029355987498554718011503310551884202013} a^{12} + \frac{9395972331069001614045301765121466015114275642498108214}{30046336950257786029355987498554718011503310551884202013} a^{11} - \frac{12934703848110494647024695338156263411912740055464755424}{30046336950257786029355987498554718011503310551884202013} a^{10} + \frac{6118053180943812046585280728614263376280743735743778493}{30046336950257786029355987498554718011503310551884202013} a^{9} + \frac{6616971581848131099566006319847148681173508634654133695}{30046336950257786029355987498554718011503310551884202013} a^{8} - \frac{3694515854528819874666587871335722090918068489366128599}{30046336950257786029355987498554718011503310551884202013} a^{7} + \frac{12253777456204812069628858379235456257694457160512286147}{30046336950257786029355987498554718011503310551884202013} a^{6} + \frac{3669086148687407201712456904750457227714684566182785939}{30046336950257786029355987498554718011503310551884202013} a^{5} + \frac{597465516850669326129327395390645884710466696236539752}{30046336950257786029355987498554718011503310551884202013} a^{4} + \frac{1579256150059602398312921124803316667619380204106826432}{30046336950257786029355987498554718011503310551884202013} a^{3} - \frac{10184624287015456621048236055321820341272629807108235327}{30046336950257786029355987498554718011503310551884202013} a^{2} + \frac{6180382714380379206742737505814886273873111837101416939}{30046336950257786029355987498554718011503310551884202013} a + \frac{13010362828572772830088519251591885996755060048822856151}{30046336950257786029355987498554718011503310551884202013}$

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:  $11$
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:  \( 638378353.767 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1869:

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 t16n1869 are not computed
Character table for t16n1869 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.8.661518125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
439Data not computed
2411Data not computed