↑

Properties

Label 16.0.16726428348...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{8}\cdot 761^{5}$
Root discriminant $50.29$
Ramified primes $2, 5, 761$
Class number $628$ (GRH)
Class group $[628]$ (GRH)
Galois group $C_2.D_4^2.C_2$ (as 16T659)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![183479, -334330, 420159, -270588, 226080, -127204, 71594, -13300, 15611, -1916, 3603, -504, 527, -56, 37, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 37*x^14 - 56*x^13 + 527*x^12 - 504*x^11 + 3603*x^10 - 1916*x^9 + 15611*x^8 - 13300*x^7 + 71594*x^6 - 127204*x^5 + 226080*x^4 - 270588*x^3 + 420159*x^2 - 334330*x + 183479)
 
gp: K = bnfinit(x^16 - 2*x^15 + 37*x^14 - 56*x^13 + 527*x^12 - 504*x^11 + 3603*x^10 - 1916*x^9 + 15611*x^8 - 13300*x^7 + 71594*x^6 - 127204*x^5 + 226080*x^4 - 270588*x^3 + 420159*x^2 - 334330*x + 183479, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 37 x^{14} - 56 x^{13} + 527 x^{12} - 504 x^{11} + 3603 x^{10} - 1916 x^{9} + 15611 x^{8} - 13300 x^{7} + 71594 x^{6} - 127204 x^{5} + 226080 x^{4} - 270588 x^{3} + 420159 x^{2} - 334330 x + 183479 \)

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:  \(1672642834856679833600000000=2^{24}\cdot 5^{8}\cdot 761^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.29$
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:  $2, 5, 761$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{292650117890619433163734412985134746807727} a^{15} - \frac{143690556446885812132967039283946982578284}{292650117890619433163734412985134746807727} a^{14} + \frac{74074585342768358552996487063415191682043}{292650117890619433163734412985134746807727} a^{13} + \frac{28384600695974958772449716416124949800220}{292650117890619433163734412985134746807727} a^{12} + \frac{60627949461669633606025808118306998083938}{292650117890619433163734412985134746807727} a^{11} + \frac{37717601779312978070879711719445684118006}{292650117890619433163734412985134746807727} a^{10} - \frac{70305452615244032273590715143866028636048}{292650117890619433163734412985134746807727} a^{9} + \frac{29358951048249987890244108684215246834667}{292650117890619433163734412985134746807727} a^{8} - \frac{119336277441392519885484246996106588548891}{292650117890619433163734412985134746807727} a^{7} + \frac{135633515193485511098272974914929366678241}{292650117890619433163734412985134746807727} a^{6} + \frac{30046161884325531774313294859549164938961}{292650117890619433163734412985134746807727} a^{5} + \frac{135372775646101882449774748842025621653197}{292650117890619433163734412985134746807727} a^{4} + \frac{81852116913350711238249406075761458670321}{292650117890619433163734412985134746807727} a^{3} - \frac{59615165071019408870446460756786876004326}{292650117890619433163734412985134746807727} a^{2} - \frac{38267766887313826285259078028165158915020}{292650117890619433163734412985134746807727} a - \frac{72308265606574337785548551845962500328643}{292650117890619433163734412985134746807727}$

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

Class group and class number

$C_{628}$, which has order $628$ (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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 9562.41394678 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.D_4^2.C_2$ (as 16T659):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 25 conjugacy class representatives for $C_2.D_4^2.C_2$
Character table for $C_2.D_4^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.1948160000.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$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}$
761Data not computed