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Properties

Label 16.0.4897760256000000.1
Degree $16$
Signature $[0, 8]$
Discriminant $4.898\times 10^{15}$
Root discriminant $9.56$
Ramified primes $2, 3, 5$
Class number $1$
Class group trivial
Galois group $(C_4\times C_8):C_2$ (as 16T114)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 6*x^13 - 10*x^12 + 28*x^11 - 8*x^10 - 90*x^9 + 253*x^8 - 396*x^7 + 433*x^6 - 352*x^5 + 218*x^4 - 102*x^3 + 35*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^14 - 6*x^13 - 10*x^12 + 28*x^11 - 8*x^10 - 90*x^9 + 253*x^8 - 396*x^7 + 433*x^6 - 352*x^5 + 218*x^4 - 102*x^3 + 35*x^2 - 8*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 35, -102, 218, -352, 433, -396, 253, -90, -8, 28, -10, -6, 8, -4, 1]);
 

\( x^{16} - 4 x^{15} + 8 x^{14} - 6 x^{13} - 10 x^{12} + 28 x^{11} - 8 x^{10} - 90 x^{9} + 253 x^{8} - 396 x^{7} + 433 x^{6} - 352 x^{5} + 218 x^{4} - 102 x^{3} + 35 x^{2} - 8 x + 1 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(4897760256000000\)\(\medspace = 2^{16}\cdot 3^{14}\cdot 5^{6}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $9.56$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 5$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{31} a^{15} - \frac{1}{31} a^{14} + \frac{5}{31} a^{13} + \frac{9}{31} a^{12} - \frac{14}{31} a^{11} - \frac{14}{31} a^{10} + \frac{12}{31} a^{9} + \frac{8}{31} a^{8} - \frac{2}{31} a^{7} + \frac{1}{31} a^{6} + \frac{2}{31} a^{5} - \frac{5}{31} a^{4} - \frac{14}{31} a^{3} + \frac{11}{31} a^{2} + \frac{6}{31} a + \frac{10}{31}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{628}{31} a^{15} - \frac{2705}{31} a^{14} + \frac{5248}{31} a^{13} - \frac{3555}{31} a^{12} - \frac{7893}{31} a^{11} + \frac{19945}{31} a^{10} - \frac{3624}{31} a^{9} - \frac{64912}{31} a^{8} + \frac{170050}{31} a^{7} - \frac{249666}{31} a^{6} + \frac{250775}{31} a^{5} - \frac{183529}{31} a^{4} + \frac{99119}{31} a^{3} - \frac{38848}{31} a^{2} + \frac{10030}{31} a - \frac{1408}{31} \) (order $12$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 59.5214550029 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 59.5214550029 \cdot 1}{12\sqrt{4897760256000000}}\approx 0.172160088985$

Galois group

$(C_4\times C_8):C_2$ (as 16T114):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_4\times C_8):C_2$
Character table for $(C_4\times C_8):C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), 8.0.4665600.1

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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
3Data not computed
$5$5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
5.8.6.4$x^{8} - 5 x^{4} + 50$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$