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Properties

Label 16.4.11984260029...8144.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{34}\cdot 17^{8}$
Root discriminant $17.99$
Ramified primes $2, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times D_8):C_2$ (as 16T126)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 16, -44, 52, -20, -16, 58, -144, 111, -36, -34, -12, 2, -4, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 4*x^13 + 2*x^12 - 12*x^11 - 34*x^10 - 36*x^9 + 111*x^8 - 144*x^7 + 58*x^6 - 16*x^5 - 20*x^4 + 52*x^3 - 44*x^2 + 16*x - 2)
 
gp: K = bnfinit(x^16 - 4*x^14 - 4*x^13 + 2*x^12 - 12*x^11 - 34*x^10 - 36*x^9 + 111*x^8 - 144*x^7 + 58*x^6 - 16*x^5 - 20*x^4 + 52*x^3 - 44*x^2 + 16*x - 2, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 4 x^{13} + 2 x^{12} - 12 x^{11} - 34 x^{10} - 36 x^{9} + 111 x^{8} - 144 x^{7} + 58 x^{6} - 16 x^{5} - 20 x^{4} + 52 x^{3} - 44 x^{2} + 16 x - 2 \)

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:  \(119842600295694598144=2^{34}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.99$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10}$, $\frac{1}{198211996398274} a^{15} - \frac{43539790455901}{198211996398274} a^{14} + \frac{19822337739669}{99105998199137} a^{13} + \frac{367912100783}{99105998199137} a^{12} - \frac{93597302822865}{198211996398274} a^{11} + \frac{22657965342915}{198211996398274} a^{10} + \frac{2766945347829}{99105998199137} a^{9} + \frac{35604353942661}{99105998199137} a^{8} - \frac{44953648292578}{99105998199137} a^{7} + \frac{8005941628398}{99105998199137} a^{6} + \frac{39512789585171}{99105998199137} a^{5} + \frac{15090040425672}{99105998199137} a^{4} - \frac{20230890147103}{99105998199137} a^{3} - \frac{47754071129183}{99105998199137} a^{2} - \frac{16175153749629}{99105998199137} a - \frac{47255114963565}{99105998199137}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 10858.3618202 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_8):C_2$ (as 16T126):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $(C_2\times D_8):C_2$
Character table for $(C_2\times D_8):C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.2.9248.1, 4.2.1156.1, 4.4.9248.1, 8.2.1368408064.1, 8.2.342102016.1, 8.4.342102016.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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

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$2.8.18.54$x^{8} + 6 x^{6} + 4 x^{3} + 6$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
2.8.16.5$x^{8} + 4 x^{6} + 40 x^{2} + 4$$4$$2$$16$$D_4$$[2, 3]^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$