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Properties

Label 16.0.63221756925...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{8}\cdot 29^{4}\cdot 1109^{3}$
Root discriminant $54.65$
Ramified primes $2, 5, 29, 1109$
Class number $186$ (GRH)
Class group $[186]$ (GRH)
Galois group 16T1741

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![35321, 66586, 22326, 41710, 4243, -37640, 66552, -16316, 3484, -378, 858, 156, 204, -96, 46, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 46*x^14 - 96*x^13 + 204*x^12 + 156*x^11 + 858*x^10 - 378*x^9 + 3484*x^8 - 16316*x^7 + 66552*x^6 - 37640*x^5 + 4243*x^4 + 41710*x^3 + 22326*x^2 + 66586*x + 35321)
 
gp: K = bnfinit(x^16 - 8*x^15 + 46*x^14 - 96*x^13 + 204*x^12 + 156*x^11 + 858*x^10 - 378*x^9 + 3484*x^8 - 16316*x^7 + 66552*x^6 - 37640*x^5 + 4243*x^4 + 41710*x^3 + 22326*x^2 + 66586*x + 35321, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 46 x^{14} - 96 x^{13} + 204 x^{12} + 156 x^{11} + 858 x^{10} - 378 x^{9} + 3484 x^{8} - 16316 x^{7} + 66552 x^{6} - 37640 x^{5} + 4243 x^{4} + 41710 x^{3} + 22326 x^{2} + 66586 x + 35321 \)

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:  \(6322175692565046886400000000=2^{24}\cdot 5^{8}\cdot 29^{4}\cdot 1109^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.65$
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:  $2, 5, 29, 1109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{16701999851560735610564153710480195443685439} a^{15} + \frac{591759382569075216655121477793324608908403}{16701999851560735610564153710480195443685439} a^{14} - \frac{6858169967979833974619305606028404444327779}{16701999851560735610564153710480195443685439} a^{13} - \frac{3953751049760430470757392796468508405434995}{16701999851560735610564153710480195443685439} a^{12} + \frac{7470407709036286948170249939907169542937021}{16701999851560735610564153710480195443685439} a^{11} + \frac{369945176621860547155787322808165272713929}{1284769219350825816197242593113861187975803} a^{10} - \frac{363573326168699713118063639566272687053652}{1284769219350825816197242593113861187975803} a^{9} - \frac{2488699473177542142616174301195151747594146}{16701999851560735610564153710480195443685439} a^{8} + \frac{204968844634706151272637160639062448125778}{1284769219350825816197242593113861187975803} a^{7} + \frac{2895250741577889704541105662933630158183134}{16701999851560735610564153710480195443685439} a^{6} - \frac{4879006020109625626031455588959438340880203}{16701999851560735610564153710480195443685439} a^{5} + \frac{8129349524377065830121060758798155746904666}{16701999851560735610564153710480195443685439} a^{4} - \frac{1148480867369382011445996093384274111447819}{16701999851560735610564153710480195443685439} a^{3} - \frac{3941212467866490008232549422803302675151741}{16701999851560735610564153710480195443685439} a^{2} + \frac{3361239122715977326205128442601719967900492}{16701999851560735610564153710480195443685439} a - \frac{462340134484011913552177211361020044744978}{1284769219350825816197242593113861187975803}$

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

Class group and class number

$C_{186}$, which has order $186$ (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:  \( 92956.9051198 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1741:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 95 conjugacy class representatives for t16n1741 are not computed
Character table for t16n1741 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.149227040000.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
5Data not computed
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
1109Data not computed