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Properties

Label 16.0.29960650073...9536.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 17^{8}$
Root discriminant $16.49$
Ramified primes $2, 17$
Class number $2$
Class group $[2]$
Galois group $D_4\times C_2$ (as 16T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -40, 172, -400, 498, -136, -510, 472, 333, -124, -2, 56, 8, -4, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 4*x^13 + 8*x^12 + 56*x^11 - 2*x^10 - 124*x^9 + 333*x^8 + 472*x^7 - 510*x^6 - 136*x^5 + 498*x^4 - 400*x^3 + 172*x^2 - 40*x + 4)
 
gp: K = bnfinit(x^16 - 4*x^14 - 4*x^13 + 8*x^12 + 56*x^11 - 2*x^10 - 124*x^9 + 333*x^8 + 472*x^7 - 510*x^6 - 136*x^5 + 498*x^4 - 400*x^3 + 172*x^2 - 40*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 4 x^{13} + 8 x^{12} + 56 x^{11} - 2 x^{10} - 124 x^{9} + 333 x^{8} + 472 x^{7} - 510 x^{6} - 136 x^{5} + 498 x^{4} - 400 x^{3} + 172 x^{2} - 40 x + 4 \)

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:  \(29960650073923649536=2^{32}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.49$
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 Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{8} - \frac{1}{2} a^{4} + \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{18} a^{11} + \frac{1}{18} a^{8} + \frac{1}{18} a^{7} - \frac{2}{9} a^{6} - \frac{1}{9} a^{5} - \frac{7}{18} a^{4} + \frac{1}{9} a^{2} + \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{18} a^{12} + \frac{1}{18} a^{9} + \frac{1}{18} a^{8} + \frac{1}{9} a^{7} - \frac{4}{9} a^{6} - \frac{1}{18} a^{5} - \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{1}{9} a^{2} - \frac{4}{9} a - \frac{1}{3}$, $\frac{1}{18} a^{13} + \frac{1}{18} a^{10} + \frac{1}{18} a^{9} - \frac{1}{18} a^{8} - \frac{1}{9} a^{7} - \frac{7}{18} a^{6} - \frac{7}{18} a^{4} + \frac{2}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3}$, $\frac{1}{13878} a^{14} - \frac{2}{2313} a^{13} + \frac{277}{13878} a^{12} - \frac{173}{6939} a^{11} - \frac{1067}{13878} a^{10} + \frac{154}{2313} a^{9} + \frac{5}{4626} a^{8} - \frac{80}{6939} a^{7} + \frac{325}{771} a^{6} - \frac{1877}{6939} a^{5} - \frac{866}{2313} a^{4} + \frac{211}{2313} a^{3} + \frac{328}{771} a^{2} + \frac{3295}{6939} a - \frac{2797}{6939}$, $\frac{1}{24716718} a^{15} + \frac{6}{457717} a^{14} + \frac{104135}{12358359} a^{13} + \frac{164326}{12358359} a^{12} + \frac{264002}{12358359} a^{11} + \frac{127241}{4119453} a^{10} + \frac{44281}{915434} a^{9} - \frac{1904887}{24716718} a^{8} - \frac{356471}{2746302} a^{7} + \frac{3411115}{12358359} a^{6} + \frac{3115939}{8238906} a^{5} - \frac{2632133}{8238906} a^{4} + \frac{900962}{4119453} a^{3} - \frac{859088}{12358359} a^{2} - \frac{2244901}{12358359} a + \frac{1913384}{4119453}$

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:  \( \frac{4165333}{8238906} a^{15} + \frac{4182689}{24716718} a^{14} - \frac{16273721}{8238906} a^{13} - \frac{33173138}{12358359} a^{12} + \frac{78711025}{24716718} a^{11} + \frac{727356299}{24716718} a^{10} + \frac{36091781}{4119453} a^{9} - \frac{248511200}{4119453} a^{8} + \frac{1830340970}{12358359} a^{7} + \frac{1192383782}{4119453} a^{6} - \frac{4043947757}{24716718} a^{5} - \frac{529306745}{4119453} a^{4} + \frac{876322771}{4119453} a^{3} - \frac{521443820}{4119453} a^{2} + \frac{514763285}{12358359} a - \frac{64830371}{12358359} \) (order $8$)
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:  \( 5277.21342233 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4$ (as 16T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(i, \sqrt{34})\), \(\Q(i, \sqrt{17})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{-2}, \sqrt{17})\), \(\Q(\sqrt{2}, \sqrt{-17})\), \(\Q(\sqrt{2}, \sqrt{17})\), \(\Q(\sqrt{-2}, \sqrt{-17})\), 4.2.2312.1 x2, 4.2.9248.1 x2, 4.0.1088.1 x2, 4.0.4352.1 x2, 8.0.5473632256.1, 8.0.342102016.3 x2, 8.0.18939904.3 x2, 8.0.342102016.4, 8.0.5473632256.5, 8.4.1368408064.2 x2, 8.0.5473632256.4 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[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}$