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Properties

Label 16.0.11715987587...5009.3
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 29^{14}$
Root discriminant $179.60$
Ramified primes $13, 29$
Class number $11664$ (GRH)
Class group $[2, 54, 108]$ (GRH)
Galois group $OD_{16}.C_2$ (as 16T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![118910117, -224991882, -48509444, 77034741, 94081513, 61121472, 29242029, 10065610, 2911429, 557525, 112504, 8963, 1722, -111, 35, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 35*x^14 - 111*x^13 + 1722*x^12 + 8963*x^11 + 112504*x^10 + 557525*x^9 + 2911429*x^8 + 10065610*x^7 + 29242029*x^6 + 61121472*x^5 + 94081513*x^4 + 77034741*x^3 - 48509444*x^2 - 224991882*x + 118910117)
 
gp: K = bnfinit(x^16 - 3*x^15 + 35*x^14 - 111*x^13 + 1722*x^12 + 8963*x^11 + 112504*x^10 + 557525*x^9 + 2911429*x^8 + 10065610*x^7 + 29242029*x^6 + 61121472*x^5 + 94081513*x^4 + 77034741*x^3 - 48509444*x^2 - 224991882*x + 118910117, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 35 x^{14} - 111 x^{13} + 1722 x^{12} + 8963 x^{11} + 112504 x^{10} + 557525 x^{9} + 2911429 x^{8} + 10065610 x^{7} + 29242029 x^{6} + 61121472 x^{5} + 94081513 x^{4} + 77034741 x^{3} - 48509444 x^{2} - 224991882 x + 118910117 \)

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:  \(1171598758708107367475386427203165009=13^{14}\cdot 29^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $179.60$
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:  $13, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{41695119692009647924318489010044368529892848148269361072356} a^{15} - \frac{72606000277527889761679914491489620051525632159205295437}{41695119692009647924318489010044368529892848148269361072356} a^{14} + \frac{231434660315403428875682460269384283995294452692907066785}{10423779923002411981079622252511092132473212037067340268089} a^{13} + \frac{1793324012355179948749617604805096905955567997721500237349}{20847559846004823962159244505022184264946424074134680536178} a^{12} + \frac{3035939694710554228898068987560851232407114213588016030335}{20847559846004823962159244505022184264946424074134680536178} a^{11} + \frac{2123946019538842817116455550114735306035472892724148522537}{10423779923002411981079622252511092132473212037067340268089} a^{10} - \frac{18806710175743900812212443946276349847773725847030041143861}{41695119692009647924318489010044368529892848148269361072356} a^{9} + \frac{3847333678031725680701685704592391983398054678632143408807}{10423779923002411981079622252511092132473212037067340268089} a^{8} - \frac{2508424970836218381827517384866052099038999413238509599347}{10423779923002411981079622252511092132473212037067340268089} a^{7} - \frac{4298794708292682819882212590640930398243041343859673588531}{20847559846004823962159244505022184264946424074134680536178} a^{6} + \frac{3728895897277572318046994804436971449075967244763758115026}{10423779923002411981079622252511092132473212037067340268089} a^{5} - \frac{12279141646154017057240875923108054632116381703186521291111}{41695119692009647924318489010044368529892848148269361072356} a^{4} - \frac{2051584298680294824349005956981226026917551573172042705168}{10423779923002411981079622252511092132473212037067340268089} a^{3} + \frac{10050080193996824810835040740862997839175691735789869845717}{20847559846004823962159244505022184264946424074134680536178} a^{2} + \frac{7050737257133287967387195201484376080502539419628633052147}{20847559846004823962159244505022184264946424074134680536178} a - \frac{4338723040043219953416811311653454389106448774319080527373}{41695119692009647924318489010044368529892848148269361072356}$

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

Class group and class number

$C_{2}\times C_{54}\times C_{108}$, which has order $11664$ (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:  \( -1 \) (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:  \( 16477435.4552 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}.C_2$ (as 16T40):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $OD_{16}.C_2$
Character table for $OD_{16}.C_2$

Intermediate fields

\(\Q(\sqrt{377}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{29}) \), 4.4.53582633.2, 4.4.53582633.1, \(\Q(\sqrt{13}, \sqrt{29})\), 8.8.2871098559212689.3

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.8.7.1$x^{8} - 13$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.1$x^{8} - 13$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$29$29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$