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Properties

Label 16.0.61870785919...4656.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{14}\cdot 7^{6}$
Root discriminant $30.69$
Ramified primes $2, 3, 7$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T486)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![603, -3132, 5904, -3888, -1782, 2916, 1620, -4176, 2364, -108, -384, 0, 114, -12, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 - 12*x^13 + 114*x^12 - 384*x^10 - 108*x^9 + 2364*x^8 - 4176*x^7 + 1620*x^6 + 2916*x^5 - 1782*x^4 - 3888*x^3 + 5904*x^2 - 3132*x + 603)
 
gp: K = bnfinit(x^16 - 12*x^14 - 12*x^13 + 114*x^12 - 384*x^10 - 108*x^9 + 2364*x^8 - 4176*x^7 + 1620*x^6 + 2916*x^5 - 1782*x^4 - 3888*x^3 + 5904*x^2 - 3132*x + 603, 1)
 

Normalized defining polynomial

\( x^{16} - 12 x^{14} - 12 x^{13} + 114 x^{12} - 384 x^{10} - 108 x^{9} + 2364 x^{8} - 4176 x^{7} + 1620 x^{6} + 2916 x^{5} - 1782 x^{4} - 3888 x^{3} + 5904 x^{2} - 3132 x + 603 \)

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:  \(618707859192665295814656=2^{40}\cdot 3^{14}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.69$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{1092} a^{14} - \frac{2}{91} a^{13} + \frac{1}{26} a^{12} + \frac{1}{26} a^{11} + \frac{23}{1092} a^{10} - \frac{15}{182} a^{9} - \frac{19}{546} a^{8} - \frac{22}{91} a^{7} + \frac{139}{364} a^{6} + \frac{19}{91} a^{5} + \frac{25}{182} a^{4} - \frac{3}{26} a^{3} - \frac{115}{364} a^{2} - \frac{45}{182} a + \frac{47}{182}$, $\frac{1}{20652098378351076} a^{15} + \frac{646513836835}{6884032792783692} a^{14} - \frac{11180842324035}{327811085370652} a^{13} + \frac{581955293533}{25216237336204} a^{12} - \frac{344490507761831}{6884032792783692} a^{11} + \frac{539209918318601}{6884032792783692} a^{10} + \frac{11348660185133}{176513661353428} a^{9} - \frac{26813765605941}{2294677597594564} a^{8} + \frac{1110134253514747}{6884032792783692} a^{7} - \frac{679771382814207}{2294677597594564} a^{6} + \frac{69315020622927}{176513661353428} a^{5} + \frac{1879752988955}{17253215019508} a^{4} - \frac{11638996092855}{38892840637196} a^{3} + \frac{997880842370767}{2294677597594564} a^{2} + \frac{483778465922445}{2294677597594564} a + \frac{104266178243901}{327811085370652}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{131004835111}{331792596529} a^{15} + \frac{251566333157}{995377789587} a^{14} - \frac{3035569682785}{663585193058} a^{13} - \frac{30511557315545}{3981511158348} a^{12} + \frac{13301198072939}{331792596529} a^{11} + \frac{16994298426933}{663585193058} a^{10} - \frac{89647799249083}{663585193058} a^{9} - \frac{171060452577865}{1327170386116} a^{8} + \frac{282200207548005}{331792596529} a^{7} - \frac{366734058539563}{331792596529} a^{6} - \frac{43065954144273}{663585193058} a^{5} + \frac{1469680494846777}{1327170386116} a^{4} + \frac{19065653535}{5623603331} a^{3} - \frac{1015565347631367}{663585193058} a^{2} + \frac{898605897245421}{663585193058} a - \frac{496548639730825}{1327170386116} \) (order $6$)
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:  \( 678431.409127 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T486):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.12096.1, 4.0.1008.2, 4.0.1728.1, 8.0.2341011456.9

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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

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
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$