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Properties

Label 16.8.93432784589...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{44}\cdot 5^{8}\cdot 17^{2}\cdot 19^{6}$
Root discriminant $64.66$
Ramified primes $2, 5, 17, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1228

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-124751, -606364, -1259640, -1464628, -1023550, -429884, -123008, -32036, 5942, 10420, 2560, 652, 242, -28, -24, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 24*x^14 - 28*x^13 + 242*x^12 + 652*x^11 + 2560*x^10 + 10420*x^9 + 5942*x^8 - 32036*x^7 - 123008*x^6 - 429884*x^5 - 1023550*x^4 - 1464628*x^3 - 1259640*x^2 - 606364*x - 124751)
 
gp: K = bnfinit(x^16 - 4*x^15 - 24*x^14 - 28*x^13 + 242*x^12 + 652*x^11 + 2560*x^10 + 10420*x^9 + 5942*x^8 - 32036*x^7 - 123008*x^6 - 429884*x^5 - 1023550*x^4 - 1464628*x^3 - 1259640*x^2 - 606364*x - 124751, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 24 x^{14} - 28 x^{13} + 242 x^{12} + 652 x^{11} + 2560 x^{10} + 10420 x^{9} + 5942 x^{8} - 32036 x^{7} - 123008 x^{6} - 429884 x^{5} - 1023550 x^{4} - 1464628 x^{3} - 1259640 x^{2} - 606364 x - 124751 \)

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:  \(93432784589729195622400000000=2^{44}\cdot 5^{8}\cdot 17^{2}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.66$
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:  $2, 5, 17, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{30} a^{14} - \frac{7}{30} a^{13} - \frac{2}{15} a^{12} + \frac{1}{5} a^{11} + \frac{1}{10} a^{10} + \frac{7}{30} a^{9} + \frac{1}{30} a^{8} + \frac{1}{30} a^{6} + \frac{1}{30} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{30} a^{2} - \frac{1}{30} a - \frac{11}{30}$, $\frac{1}{2328287920204281596643251322702014730} a^{15} + \frac{7429325136245596304940739486826141}{2328287920204281596643251322702014730} a^{14} + \frac{22933993750349018407729500663365150}{232828792020428159664325132270201473} a^{13} + \frac{9723351030736528992478700479496813}{776095973401427198881083774234004910} a^{12} + \frac{4463532542732903056863591264246201}{35277089700064872676412898828818405} a^{11} - \frac{14863552481576779960620531076816777}{1164143960102140798321625661351007365} a^{10} - \frac{223070988918866328370382228863133749}{1164143960102140798321625661351007365} a^{9} - \frac{77187617704427575714411936842990249}{776095973401427198881083774234004910} a^{8} - \frac{228171448787116145105189966072786909}{2328287920204281596643251322702014730} a^{7} - \frac{266922165823273734765615484922401751}{2328287920204281596643251322702014730} a^{6} + \frac{147198805198366212927233979362763286}{388047986700713599440541887117002455} a^{5} - \frac{51676195825148294158835178979166621}{155219194680285439776216754846800982} a^{4} - \frac{50107546467959608773212100021358429}{105831269100194618029238696486455215} a^{3} + \frac{32584816171502590692272752294474853}{105831269100194618029238696486455215} a^{2} - \frac{447566829914795111009347872498970027}{1164143960102140798321625661351007365} a - \frac{6215698350542289523776737694140931}{70554179400129745352825797657636810}$

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:  \( 459858409.514 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1228:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1228 are not computed
Character table for t16n1228 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.4.7600.1, 4.4.30400.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.14786560000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.2.0.1$x^{2} - x + 3$$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}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.8.6.1$x^{8} + 57 x^{4} + 1444$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$