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Properties

Label 16.8.14009685799...5625.2
Degree $16$
Signature $[8, 4]$
Discriminant $3^{8}\cdot 5^{10}\cdot 7^{8}\cdot 41^{14}$
Root discriminant $322.96$
Ramified primes $3, 5, 7, 41$
Class number $64$ (GRH)
Class group $[2, 4, 8]$ (GRH)
Galois group 16T813

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-791780750, 1880620875, -599158350, -484341650, 19776230, 54157635, -16940841, -11815071, 2984096, 1162594, -216639, -51539, 7883, 1052, -142, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 142*x^14 + 1052*x^13 + 7883*x^12 - 51539*x^11 - 216639*x^10 + 1162594*x^9 + 2984096*x^8 - 11815071*x^7 - 16940841*x^6 + 54157635*x^5 + 19776230*x^4 - 484341650*x^3 - 599158350*x^2 + 1880620875*x - 791780750)
 
gp: K = bnfinit(x^16 - 8*x^15 - 142*x^14 + 1052*x^13 + 7883*x^12 - 51539*x^11 - 216639*x^10 + 1162594*x^9 + 2984096*x^8 - 11815071*x^7 - 16940841*x^6 + 54157635*x^5 + 19776230*x^4 - 484341650*x^3 - 599158350*x^2 + 1880620875*x - 791780750, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 142 x^{14} + 1052 x^{13} + 7883 x^{12} - 51539 x^{11} - 216639 x^{10} + 1162594 x^{9} + 2984096 x^{8} - 11815071 x^{7} - 16940841 x^{6} + 54157635 x^{5} + 19776230 x^{4} - 484341650 x^{3} - 599158350 x^{2} + 1880620875 x - 791780750 \)

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:  \(14009685799460036277781846932413291015625=3^{8}\cdot 5^{10}\cdot 7^{8}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $322.96$
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:  $3, 5, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{1}{10} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{2} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{9} - \frac{1}{5} a^{7} - \frac{3}{10} a^{6} + \frac{2}{5} a^{5} + \frac{1}{10} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{10} - \frac{1}{2} a^{7} + \frac{1}{5} a^{6} - \frac{3}{10} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{3}{10} a^{2}$, $\frac{1}{20} a^{11} - \frac{1}{20} a^{10} - \frac{1}{20} a^{9} - \frac{1}{20} a^{8} - \frac{1}{4} a^{7} + \frac{1}{5} a^{6} - \frac{1}{20} a^{5} - \frac{9}{20} a^{4} - \frac{1}{4} a^{3} + \frac{1}{20} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{200} a^{12} - \frac{1}{25} a^{10} + \frac{1}{25} a^{9} - \frac{1}{20} a^{8} + \frac{93}{200} a^{7} - \frac{11}{40} a^{6} - \frac{1}{50} a^{5} - \frac{9}{50} a^{4} - \frac{2}{5} a^{3} + \frac{3}{20} a^{2} + \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{800} a^{13} - \frac{1}{800} a^{12} - \frac{1}{100} a^{11} - \frac{1}{200} a^{10} + \frac{1}{400} a^{9} - \frac{17}{800} a^{8} - \frac{23}{50} a^{7} + \frac{271}{800} a^{6} + \frac{37}{200} a^{5} + \frac{3}{25} a^{4} - \frac{9}{80} a^{3} + \frac{61}{160} a^{2} - \frac{13}{32} a + \frac{3}{16}$, $\frac{1}{11200} a^{14} - \frac{1}{2800} a^{13} + \frac{19}{11200} a^{12} - \frac{1}{80} a^{11} - \frac{89}{5600} a^{10} + \frac{249}{11200} a^{9} - \frac{11}{1600} a^{8} + \frac{721}{1600} a^{7} + \frac{167}{448} a^{6} - \frac{1191}{2800} a^{5} + \frac{1979}{5600} a^{4} - \frac{877}{2240} a^{3} + \frac{11}{280} a^{2} - \frac{59}{448} a - \frac{1}{224}$, $\frac{1}{782350129526870146668800811462212600173166356652800} a^{15} - \frac{13722922474657154532895304097884660657368747641}{782350129526870146668800811462212600173166356652800} a^{14} + \frac{370226730705715066272728101305860306465113150091}{782350129526870146668800811462212600173166356652800} a^{13} - \frac{286846929381148911055399368769202993946630537843}{156470025905374029333760162292442520034633271330560} a^{12} - \frac{1006160861409563857938067356982679849501828008943}{78235012952687014666880081146221260017316635665280} a^{11} - \frac{33190185549322053061154597245119240563699473091997}{782350129526870146668800811462212600173166356652800} a^{10} - \frac{15000115382349535991747131296475566180344135897833}{391175064763435073334400405731106300086583178326400} a^{9} + \frac{1318637407038838478321320928718379699333956851591}{27941076054531076666742886123650450006184512737600} a^{8} - \frac{15099494719328099280206293043260527312659885476097}{39117506476343507333440040573110630008658317832640} a^{7} + \frac{7143516692871682060100976607029558257799853170445}{31294005181074805866752032458488504006926654266112} a^{6} - \frac{34855745784148715157032355241852093794873905833531}{78235012952687014666880081146221260017316635665280} a^{5} + \frac{349332004607290009054034751196217923808002768435441}{782350129526870146668800811462212600173166356652800} a^{4} - \frac{800659549592803976284766105847316110414605060629}{4470572168724972266678861779784072000989522038016} a^{3} + \frac{1992972788217924017280515686634254021953753873255}{4470572168724972266678861779784072000989522038016} a^{2} + \frac{2128227011122265117072572483860121498694273492609}{31294005181074805866752032458488504006926654266112} a - \frac{311069866832387374641467293645115834878249277145}{680304460458147953625044183880184869715796831872}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{8}$, which has order $64$ (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:  \( 115530061950000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T813:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n813
Character table for t16n813 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{4305}) \), \(\Q(\sqrt{105}) \), 4.4.16885645.1, 4.4.3101445.1, \(\Q(\sqrt{41}, \sqrt{105})\), 8.8.577378139308700625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
41Data not computed