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Properties

Label 16.0.95735899582...8944.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{54}\cdot 3^{12}$
Root discriminant $23.65$
Ramified primes $2, 3$
Class number $2$
Class group $[2]$
Galois group $(C_2^2\times C_4):C_2$ (as 16T54)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![46, -16, -192, 672, 1652, 768, 344, 104, -222, 328, -136, -96, 152, -96, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 96*x^13 + 152*x^12 - 96*x^11 - 136*x^10 + 328*x^9 - 222*x^8 + 104*x^7 + 344*x^6 + 768*x^5 + 1652*x^4 + 672*x^3 - 192*x^2 - 16*x + 46)
 
gp: K = bnfinit(x^16 - 8*x^15 + 36*x^14 - 96*x^13 + 152*x^12 - 96*x^11 - 136*x^10 + 328*x^9 - 222*x^8 + 104*x^7 + 344*x^6 + 768*x^5 + 1652*x^4 + 672*x^3 - 192*x^2 - 16*x + 46, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 36 x^{14} - 96 x^{13} + 152 x^{12} - 96 x^{11} - 136 x^{10} + 328 x^{9} - 222 x^{8} + 104 x^{7} + 344 x^{6} + 768 x^{5} + 1652 x^{4} + 672 x^{3} - 192 x^{2} - 16 x + 46 \)

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:  \(9573589958277615058944=2^{54}\cdot 3^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.65$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{115} a^{14} - \frac{9}{115} a^{13} + \frac{39}{115} a^{12} + \frac{57}{115} a^{11} + \frac{22}{115} a^{10} + \frac{1}{5} a^{9} + \frac{8}{115} a^{8} - \frac{48}{115} a^{7} - \frac{38}{115} a^{6} - \frac{7}{115} a^{5} - \frac{42}{115} a^{4} + \frac{47}{115} a^{3} - \frac{29}{115} a^{2} - \frac{18}{115} a - \frac{2}{5}$, $\frac{1}{94081372890106444585} a^{15} + \frac{179542071615132036}{94081372890106444585} a^{14} + \frac{371609856463712398}{4090494473482888895} a^{13} - \frac{43003174307332812723}{94081372890106444585} a^{12} - \frac{46094241078211104668}{94081372890106444585} a^{11} - \frac{9850383132886221072}{94081372890106444585} a^{10} - \frac{19996193693589900297}{94081372890106444585} a^{9} - \frac{132839384807133426}{1288785930001458145} a^{8} - \frac{34604037120509224458}{94081372890106444585} a^{7} - \frac{11007136284305161182}{94081372890106444585} a^{6} + \frac{17997054135941774703}{94081372890106444585} a^{5} - \frac{23819656504775121178}{94081372890106444585} a^{4} + \frac{2410624032497162131}{94081372890106444585} a^{3} - \frac{43598711490110063168}{94081372890106444585} a^{2} - \frac{23815780840654537356}{94081372890106444585} a + \frac{337039196921870652}{818098894696577779}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 25742.4926099 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times C_4):C_2$ (as 16T54):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $(C_2^2\times C_4):C_2$
Character table for $(C_2^2\times C_4):C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.0.27648.1 x2, \(\Q(\sqrt{2}, \sqrt{3})\), 4.0.13824.1 x2, 8.0.3057647616.7, 8.4.1358954496.1, 8.4.12230590464.1

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

Sibling fields

Galois closure: data not computed
Degree 16 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 R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
3Data not computed