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Properties

Label 16.4.12383987090...8352.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{8}\cdot 13^{2}\cdot 17^{15}$
Root discriminant $27.75$
Ramified primes $2, 13, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1648, 7456, -16968, 27112, -32024, 29212, -21088, 12126, -5781, 2421, -950, 360, -106, 11, 8, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 8*x^14 + 11*x^13 - 106*x^12 + 360*x^11 - 950*x^10 + 2421*x^9 - 5781*x^8 + 12126*x^7 - 21088*x^6 + 29212*x^5 - 32024*x^4 + 27112*x^3 - 16968*x^2 + 7456*x - 1648)
 
gp: K = bnfinit(x^16 - 5*x^15 + 8*x^14 + 11*x^13 - 106*x^12 + 360*x^11 - 950*x^10 + 2421*x^9 - 5781*x^8 + 12126*x^7 - 21088*x^6 + 29212*x^5 - 32024*x^4 + 27112*x^3 - 16968*x^2 + 7456*x - 1648, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 8 x^{14} + 11 x^{13} - 106 x^{12} + 360 x^{11} - 950 x^{10} + 2421 x^{9} - 5781 x^{8} + 12126 x^{7} - 21088 x^{6} + 29212 x^{5} - 32024 x^{4} + 27112 x^{3} - 16968 x^{2} + 7456 x - 1648 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(123839870900520670468352=2^{8}\cdot 13^{2}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.75$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{536} a^{14} - \frac{9}{536} a^{13} + \frac{31}{268} a^{12} - \frac{131}{536} a^{11} - \frac{37}{268} a^{10} + \frac{87}{268} a^{9} - \frac{15}{268} a^{8} + \frac{45}{536} a^{7} - \frac{69}{536} a^{6} - \frac{47}{134} a^{5} + \frac{65}{268} a^{4} + \frac{29}{134} a^{3} + \frac{17}{67} a^{2} + \frac{31}{67} a + \frac{4}{67}$, $\frac{1}{4268782460961261368} a^{15} + \frac{1487934911985699}{4268782460961261368} a^{14} + \frac{82342461857362863}{2134391230480630684} a^{13} - \frac{271560705600654389}{4268782460961261368} a^{12} - \frac{35984391550338667}{533597807620157671} a^{11} - \frac{438935796208502953}{2134391230480630684} a^{10} - \frac{186390459030199587}{1067195615240315342} a^{9} - \frac{1073717315695140391}{4268782460961261368} a^{8} - \frac{1381028960971377789}{4268782460961261368} a^{7} - \frac{185195044288438003}{533597807620157671} a^{6} - \frac{155014295365553782}{533597807620157671} a^{5} + \frac{709089617514323313}{2134391230480630684} a^{4} + \frac{108117479321725638}{533597807620157671} a^{3} - \frac{472845996683605113}{1067195615240315342} a^{2} - \frac{62047723364489177}{533597807620157671} a - \frac{101616883521050}{5180561239030657}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 71184.5952122 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

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 $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 sibling: 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 $16$ $16$ $16$ $16$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17Data not computed