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Properties

Label 16.8.56148868668...6129.8
Degree $16$
Signature $[8, 4]$
Discriminant $23^{6}\cdot 41^{14}$
Root discriminant $83.53$
Ramified primes $23, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2822581, -5491088, 4472119, -1074064, -1448971, 1430580, -365052, -131446, 102421, -16974, -2430, 332, 124, 88, -30, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 30*x^14 + 88*x^13 + 124*x^12 + 332*x^11 - 2430*x^10 - 16974*x^9 + 102421*x^8 - 131446*x^7 - 365052*x^6 + 1430580*x^5 - 1448971*x^4 - 1074064*x^3 + 4472119*x^2 - 5491088*x + 2822581)
 
gp: K = bnfinit(x^16 - 2*x^15 - 30*x^14 + 88*x^13 + 124*x^12 + 332*x^11 - 2430*x^10 - 16974*x^9 + 102421*x^8 - 131446*x^7 - 365052*x^6 + 1430580*x^5 - 1448971*x^4 - 1074064*x^3 + 4472119*x^2 - 5491088*x + 2822581, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 30 x^{14} + 88 x^{13} + 124 x^{12} + 332 x^{11} - 2430 x^{10} - 16974 x^{9} + 102421 x^{8} - 131446 x^{7} - 365052 x^{6} + 1430580 x^{5} - 1448971 x^{4} - 1074064 x^{3} + 4472119 x^{2} - 5491088 x + 2822581 \)

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:  \(5614886866882301027209678756129=23^{6}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.53$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{118} a^{14} + \frac{3}{59} a^{13} + \frac{23}{118} a^{12} + \frac{7}{118} a^{11} + \frac{13}{118} a^{9} - \frac{25}{118} a^{8} - \frac{29}{59} a^{7} + \frac{57}{118} a^{6} - \frac{5}{118} a^{5} - \frac{5}{59} a^{4} - \frac{39}{118} a^{3} - \frac{57}{118} a^{2} - \frac{15}{59} a + \frac{43}{118}$, $\frac{1}{8877327409417961841508698320995034001302} a^{15} - \frac{131243093720777874246334679611614061}{143182700151902610346914489048307000021} a^{14} - \frac{124447795365185878266600151284190487085}{8877327409417961841508698320995034001302} a^{13} + \frac{681084196221246754111458440026055248689}{4438663704708980920754349160497517000651} a^{12} - \frac{1676269181038856850394594042137238560881}{8877327409417961841508698320995034001302} a^{11} - \frac{1460455542371920040911698603513924014213}{8877327409417961841508698320995034001302} a^{10} - \frac{385363906251260969331802576537869203393}{8877327409417961841508698320995034001302} a^{9} - \frac{924540671061214790841280363305076053875}{8877327409417961841508698320995034001302} a^{8} + \frac{675083259884094242717397505770960802886}{4438663704708980920754349160497517000651} a^{7} - \frac{972327757998681120060390541328135738323}{4438663704708980920754349160497517000651} a^{6} - \frac{296083542743186563876662248893910597408}{4438663704708980920754349160497517000651} a^{5} - \frac{2396048416311884494336385989727140073731}{8877327409417961841508698320995034001302} a^{4} - \frac{119080751434740618483392634684633091690}{4438663704708980920754349160497517000651} a^{3} - \frac{2161934131185878369027887962719690944567}{4438663704708980920754349160497517000651} a^{2} - \frac{981555755836050054035382989656488258633}{8877327409417961841508698320995034001302} a + \frac{39487876782726856229217207732315290}{130522060302554795211407920736834093}$

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:  \( 4121906138.16 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.4.103025010883049.3, 8.6.57794518300247.1, 8.6.4479348299263.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$