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Properties

Label 16.4.10787325443...625.14
Degree $16$
Signature $[4, 6]$
Discriminant $5^{10}\cdot 101^{10}$
Root discriminant $48.93$
Ramified primes $5, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T875

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5125, -16050, 1696, 20962, 819, -3912, -5494, 947, 1574, -582, 381, -368, 176, -63, 24, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 24*x^14 - 63*x^13 + 176*x^12 - 368*x^11 + 381*x^10 - 582*x^9 + 1574*x^8 + 947*x^7 - 5494*x^6 - 3912*x^5 + 819*x^4 + 20962*x^3 + 1696*x^2 - 16050*x - 5125)
 
gp: K = bnfinit(x^16 - 7*x^15 + 24*x^14 - 63*x^13 + 176*x^12 - 368*x^11 + 381*x^10 - 582*x^9 + 1574*x^8 + 947*x^7 - 5494*x^6 - 3912*x^5 + 819*x^4 + 20962*x^3 + 1696*x^2 - 16050*x - 5125, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 24 x^{14} - 63 x^{13} + 176 x^{12} - 368 x^{11} + 381 x^{10} - 582 x^{9} + 1574 x^{8} + 947 x^{7} - 5494 x^{6} - 3912 x^{5} + 819 x^{4} + 20962 x^{3} + 1696 x^{2} - 16050 x - 5125 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1078732544346879404306640625=5^{10}\cdot 101^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.93$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{6} + \frac{2}{5} a$, $\frac{1}{50} a^{12} - \frac{1}{25} a^{11} + \frac{3}{50} a^{9} - \frac{1}{25} a^{8} + \frac{2}{25} a^{7} - \frac{7}{50} a^{6} + \frac{1}{5} a^{5} + \frac{9}{25} a^{4} - \frac{17}{50} a^{3} + \frac{9}{25} a^{2} - \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{350} a^{13} + \frac{1}{350} a^{12} + \frac{17}{175} a^{11} - \frac{1}{50} a^{10} - \frac{23}{350} a^{9} - \frac{6}{175} a^{8} + \frac{3}{70} a^{7} - \frac{1}{350} a^{6} - \frac{1}{175} a^{5} - \frac{3}{350} a^{4} - \frac{33}{350} a^{3} + \frac{17}{175} a^{2} - \frac{31}{70} a + \frac{5}{14}$, $\frac{1}{2450} a^{14} + \frac{3}{2450} a^{13} - \frac{2}{245} a^{12} + \frac{103}{2450} a^{11} - \frac{107}{2450} a^{10} + \frac{27}{1225} a^{9} - \frac{177}{2450} a^{8} - \frac{5}{98} a^{7} + \frac{194}{1225} a^{6} + \frac{79}{350} a^{5} - \frac{207}{2450} a^{4} + \frac{43}{245} a^{3} + \frac{187}{490} a^{2} + \frac{159}{490} a - \frac{9}{49}$, $\frac{1}{813110971685125835750} a^{15} + \frac{30638541624981961}{406555485842562917875} a^{14} + \frac{65273649993238158}{58079355120366131125} a^{13} + \frac{24644377169813509}{3318820292592350350} a^{12} - \frac{26104947575190791562}{406555485842562917875} a^{11} - \frac{30630178912338980152}{406555485842562917875} a^{10} - \frac{2019832091243925411}{32524438867405033430} a^{9} - \frac{28803477632631069506}{406555485842562917875} a^{8} + \frac{31389689046905959608}{406555485842562917875} a^{7} + \frac{400895545302178695901}{813110971685125835750} a^{6} - \frac{38518985640545185733}{81311097168512583575} a^{5} - \frac{37396272548258247861}{406555485842562917875} a^{4} + \frac{99254278472490151601}{813110971685125835750} a^{3} - \frac{36207731851225859807}{406555485842562917875} a^{2} + \frac{1016704310688611159}{16262219433702516715} a - \frac{52601363977238669}{3252443886740503343}$

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:  $9$
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:  \( 70265232.5616 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T875:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n875
Character table for t16n875 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.4.65037750625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
101Data not computed