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Properties

Label 16.8.18077695512...7776.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{56}\cdot 7^{8}\cdot 17^{2}\cdot 41^{4}\cdot 73^{2}$
Root discriminant $184.53$
Ramified primes $2, 7, 17, 41, 73$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T1086

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1804313, 10924760, -16868624, 435432, 6613892, -603568, -1325160, 324776, 308810, -108104, -33864, 2008, 532, 96, -40, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 40*x^14 + 96*x^13 + 532*x^12 + 2008*x^11 - 33864*x^10 - 108104*x^9 + 308810*x^8 + 324776*x^7 - 1325160*x^6 - 603568*x^5 + 6613892*x^4 + 435432*x^3 - 16868624*x^2 + 10924760*x - 1804313)
 
gp: K = bnfinit(x^16 - 8*x^15 - 40*x^14 + 96*x^13 + 532*x^12 + 2008*x^11 - 33864*x^10 - 108104*x^9 + 308810*x^8 + 324776*x^7 - 1325160*x^6 - 603568*x^5 + 6613892*x^4 + 435432*x^3 - 16868624*x^2 + 10924760*x - 1804313, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 40 x^{14} + 96 x^{13} + 532 x^{12} + 2008 x^{11} - 33864 x^{10} - 108104 x^{9} + 308810 x^{8} + 324776 x^{7} - 1325160 x^{6} - 603568 x^{5} + 6613892 x^{4} + 435432 x^{3} - 16868624 x^{2} + 10924760 x - 1804313 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1807769551224448500316223474054987776=2^{56}\cdot 7^{8}\cdot 17^{2}\cdot 41^{4}\cdot 73^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $184.53$
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:  $2, 7, 17, 41, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} + \frac{1}{7} a^{9} + \frac{2}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{9} + \frac{2}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{6} + \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{3} + \frac{3}{7} a^{2}$, $\frac{1}{49} a^{12} - \frac{2}{49} a^{11} + \frac{1}{49} a^{10} + \frac{6}{49} a^{9} - \frac{17}{49} a^{8} - \frac{9}{49} a^{7} + \frac{1}{49} a^{6} + \frac{2}{49} a^{5} + \frac{1}{7} a^{4} - \frac{9}{49} a^{3} + \frac{9}{49} a^{2} - \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{343} a^{13} + \frac{4}{343} a^{11} - \frac{20}{343} a^{10} - \frac{138}{343} a^{9} + \frac{118}{343} a^{8} - \frac{17}{343} a^{7} + \frac{39}{343} a^{6} + \frac{46}{343} a^{5} + \frac{96}{343} a^{4} - \frac{44}{343} a^{3} - \frac{80}{343} a^{2} - \frac{22}{49} a - \frac{19}{49}$, $\frac{1}{40817} a^{14} + \frac{16}{40817} a^{13} + \frac{200}{40817} a^{12} + \frac{2592}{40817} a^{11} - \frac{1046}{40817} a^{10} + \frac{164}{40817} a^{9} + \frac{4076}{40817} a^{8} + \frac{1090}{40817} a^{7} + \frac{956}{2401} a^{6} + \frac{15581}{40817} a^{5} - \frac{20313}{40817} a^{4} - \frac{345}{833} a^{3} - \frac{9764}{40817} a^{2} + \frac{234}{833} a + \frac{1166}{5831}$, $\frac{1}{100946688689962907055605409271519171643293690289} a^{15} + \frac{1397010025611386446543846987149895185622}{100946688689962907055605409271519171643293690289} a^{14} + \frac{41655273163501720574209468325183690994182709}{100946688689962907055605409271519171643293690289} a^{13} + \frac{49965523750900488288826124367193819024191521}{5938040511174288650329729957148186567252570017} a^{12} + \frac{3550144448979467037887282670397861229669645510}{100946688689962907055605409271519171643293690289} a^{11} + \frac{2544014735960334930707344740220929861535678504}{100946688689962907055605409271519171643293690289} a^{10} - \frac{46932346353134568091454815373216267036361145858}{100946688689962907055605409271519171643293690289} a^{9} - \frac{10659001970051526045514162184840480316557118423}{100946688689962907055605409271519171643293690289} a^{8} - \frac{26098021664802589952177347533075341640612093032}{100946688689962907055605409271519171643293690289} a^{7} - \frac{33621146358785916832979055223463087835759364100}{100946688689962907055605409271519171643293690289} a^{6} - \frac{3555302664486089320876320758189116810696577710}{100946688689962907055605409271519171643293690289} a^{5} - \frac{14762639002158268555610917150628015706135189567}{100946688689962907055605409271519171643293690289} a^{4} - \frac{2250527772463561513897917307323673817086531770}{100946688689962907055605409271519171643293690289} a^{3} - \frac{18134500475433647367068616660393579749828989168}{100946688689962907055605409271519171643293690289} a^{2} + \frac{4719948482309903282826196307180442790719639165}{14420955527137558150800772753074167377613384327} a + \frac{5530374447084769868401573584372616611761765525}{14420955527137558150800772753074167377613384327}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $11$
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:  \( 2795208933900 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1086:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 97 conjugacy class representatives for t16n1086 are not computed
Character table for t16n1086 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.587776.2, 4.4.100352.1, 4.4.293888.2, 8.8.270856810921984.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$