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Properties

Label 16.0.43930741219...2077.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{9}\cdot 23^{10}$
Root discriminant $30.04$
Ramified primes $13, 23$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2887, -4770, -1956, 8553, -1846, -6458, 4887, -773, -744, 941, -561, 111, 46, -44, 22, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 22*x^14 - 44*x^13 + 46*x^12 + 111*x^11 - 561*x^10 + 941*x^9 - 744*x^8 - 773*x^7 + 4887*x^6 - 6458*x^5 - 1846*x^4 + 8553*x^3 - 1956*x^2 - 4770*x + 2887)
 
gp: K = bnfinit(x^16 - 7*x^15 + 22*x^14 - 44*x^13 + 46*x^12 + 111*x^11 - 561*x^10 + 941*x^9 - 744*x^8 - 773*x^7 + 4887*x^6 - 6458*x^5 - 1846*x^4 + 8553*x^3 - 1956*x^2 - 4770*x + 2887, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 22 x^{14} - 44 x^{13} + 46 x^{12} + 111 x^{11} - 561 x^{10} + 941 x^{9} - 744 x^{8} - 773 x^{7} + 4887 x^{6} - 6458 x^{5} - 1846 x^{4} + 8553 x^{3} - 1956 x^{2} - 4770 x + 2887 \)

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:  \(439307412190718289542077=13^{9}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.04$
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:  $13, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{34} a^{13} + \frac{3}{17} a^{12} - \frac{3}{34} a^{11} + \frac{1}{17} a^{10} - \frac{4}{17} a^{9} + \frac{2}{17} a^{8} + \frac{2}{17} a^{7} - \frac{11}{34} a^{6} - \frac{1}{2} a^{5} - \frac{7}{34} a^{4} + \frac{2}{17} a^{3} - \frac{7}{34} a^{2} - \frac{11}{34} a - \frac{4}{17}$, $\frac{1}{34} a^{14} - \frac{5}{34} a^{12} + \frac{3}{34} a^{11} - \frac{3}{34} a^{10} + \frac{1}{34} a^{9} + \frac{7}{17} a^{8} + \frac{8}{17} a^{7} - \frac{1}{17} a^{6} - \frac{7}{34} a^{5} + \frac{6}{17} a^{4} - \frac{7}{17} a^{3} - \frac{3}{34} a^{2} + \frac{7}{34} a + \frac{7}{17}$, $\frac{1}{2027827706344313054} a^{15} + \frac{13873477198805291}{1013913853172156527} a^{14} + \frac{18751182945240713}{2027827706344313054} a^{13} - \frac{51207476782173333}{1013913853172156527} a^{12} - \frac{208579954423534307}{2027827706344313054} a^{11} + \frac{112722229121230765}{2027827706344313054} a^{10} - \frac{27004862830254209}{119283982726136062} a^{9} - \frac{358089234202327877}{2027827706344313054} a^{8} + \frac{15615690810319333}{1013913853172156527} a^{7} + \frac{146049273791220776}{1013913853172156527} a^{6} - \frac{167990191676322789}{1013913853172156527} a^{5} + \frac{646316553086511273}{2027827706344313054} a^{4} - \frac{328807588936308820}{1013913853172156527} a^{3} - \frac{12230400658496825}{24729606174930647} a^{2} + \frac{170979241475333863}{2027827706344313054} a + \frac{56203941772499212}{1013913853172156527}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 30590.2605268 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-23}) \), 4.0.6877.1, 8.0.614810677.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ $16$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ $16$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$13$13.4.3.4$x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$23$23.8.6.1$x^{8} + 299 x^{4} + 25921$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$