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Properties

Label 16.2.73589398538...6875.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,3^{8}\cdot 5^{14}\cdot 179^{5}$
Root discriminant $35.82$
Ramified primes $3, 5, 179$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1354

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3001, 4027, -6870, -21985, -30590, -27119, -14958, -7600, -2520, -1435, -93, -161, 55, -25, 15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 15*x^14 - 25*x^13 + 55*x^12 - 161*x^11 - 93*x^10 - 1435*x^9 - 2520*x^8 - 7600*x^7 - 14958*x^6 - 27119*x^5 - 30590*x^4 - 21985*x^3 - 6870*x^2 + 4027*x + 3001)
 
gp: K = bnfinit(x^16 - 2*x^15 + 15*x^14 - 25*x^13 + 55*x^12 - 161*x^11 - 93*x^10 - 1435*x^9 - 2520*x^8 - 7600*x^7 - 14958*x^6 - 27119*x^5 - 30590*x^4 - 21985*x^3 - 6870*x^2 + 4027*x + 3001, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 15 x^{14} - 25 x^{13} + 55 x^{12} - 161 x^{11} - 93 x^{10} - 1435 x^{9} - 2520 x^{8} - 7600 x^{7} - 14958 x^{6} - 27119 x^{5} - 30590 x^{4} - 21985 x^{3} - 6870 x^{2} + 4027 x + 3001 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-7358939853847283935546875=-\,3^{8}\cdot 5^{14}\cdot 179^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.82$
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:  $3, 5, 179$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{5} - \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{6} - \frac{1}{5} a$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{7} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{145} a^{14} + \frac{4}{145} a^{13} - \frac{6}{145} a^{12} - \frac{9}{145} a^{11} + \frac{2}{29} a^{10} + \frac{14}{145} a^{9} + \frac{1}{29} a^{8} - \frac{34}{145} a^{7} - \frac{4}{29} a^{6} - \frac{42}{145} a^{5} - \frac{42}{145} a^{4} + \frac{39}{145} a^{3} + \frac{12}{145} a^{2} - \frac{4}{145} a + \frac{19}{145}$, $\frac{1}{149608087812280041211841395} a^{15} + \frac{250231686348348949062529}{149608087812280041211841395} a^{14} - \frac{8970880258019143678443374}{149608087812280041211841395} a^{13} + \frac{222338729115049875859880}{29921617562456008242368279} a^{12} + \frac{8261223035010341301399722}{149608087812280041211841395} a^{11} + \frac{8150080949823938742899166}{149608087812280041211841395} a^{10} - \frac{4541986502750222143247014}{149608087812280041211841395} a^{9} - \frac{1422652130704151885162624}{29921617562456008242368279} a^{8} + \frac{131682260656916413369328}{149608087812280041211841395} a^{7} + \frac{39578903665234523504054373}{149608087812280041211841395} a^{6} + \frac{11791824069810374117512916}{149608087812280041211841395} a^{5} + \frac{53717327136495029781310367}{149608087812280041211841395} a^{4} + \frac{53712552312229417832820936}{149608087812280041211841395} a^{3} - \frac{6762318343577142468553266}{149608087812280041211841395} a^{2} - \frac{70122887712033039056344179}{149608087812280041211841395} a + \frac{23097947457222977693829771}{149608087812280041211841395}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $8$
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:  \( 188036.164759 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1354:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 59 conjugacy class representatives for t16n1354 are not computed
Character table for t16n1354 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.6.226546875.1

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 $16$ R R $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$179$$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
179.2.1.1$x^{2} - 179$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.1.1$x^{2} - 179$$2$$1$$1$$C_2$$[\ ]_{2}$
179.4.3.2$x^{4} - 179$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$