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Properties

Label 16.8.14009685799...5625.1
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![3013600, -36260400, 73641130, 153830565, -48085035, -11279715, -3949556, 2069598, -246448, -48585, 13226, 9471, 2908, -246, -114, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 114*x^14 - 246*x^13 + 2908*x^12 + 9471*x^11 + 13226*x^10 - 48585*x^9 - 246448*x^8 + 2069598*x^7 - 3949556*x^6 - 11279715*x^5 - 48085035*x^4 + 153830565*x^3 + 73641130*x^2 - 36260400*x + 3013600)
 
gp: K = bnfinit(x^16 - 114*x^14 - 246*x^13 + 2908*x^12 + 9471*x^11 + 13226*x^10 - 48585*x^9 - 246448*x^8 + 2069598*x^7 - 3949556*x^6 - 11279715*x^5 - 48085035*x^4 + 153830565*x^3 + 73641130*x^2 - 36260400*x + 3013600, 1)
 

Normalized defining polynomial

\( x^{16} - 114 x^{14} - 246 x^{13} + 2908 x^{12} + 9471 x^{11} + 13226 x^{10} - 48585 x^{9} - 246448 x^{8} + 2069598 x^{7} - 3949556 x^{6} - 11279715 x^{5} - 48085035 x^{4} + 153830565 x^{3} + 73641130 x^{2} - 36260400 x + 3013600 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{3}{10} a^{6} + \frac{3}{10} a^{5} - \frac{2}{5} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{60} a^{11} - \frac{1}{30} a^{10} - \frac{1}{5} a^{9} + \frac{1}{10} a^{8} - \frac{17}{60} a^{7} + \frac{13}{60} a^{6} + \frac{1}{10} a^{5} + \frac{1}{4} a^{4} - \frac{5}{12} a^{3} - \frac{5}{12} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{240} a^{12} - \frac{1}{240} a^{11} - \frac{1}{30} a^{10} - \frac{3}{40} a^{9} + \frac{37}{240} a^{8} + \frac{23}{60} a^{7} + \frac{97}{240} a^{6} + \frac{33}{80} a^{5} + \frac{43}{120} a^{4} + \frac{1}{6} a^{3} - \frac{7}{48} a^{2} - \frac{11}{24} a - \frac{1}{6}$, $\frac{1}{720} a^{13} + \frac{1}{720} a^{12} + \frac{1}{360} a^{11} + \frac{7}{360} a^{10} - \frac{47}{720} a^{9} - \frac{73}{360} a^{8} - \frac{211}{720} a^{7} - \frac{11}{144} a^{6} - \frac{37}{180} a^{5} + \frac{13}{90} a^{4} + \frac{23}{48} a^{3} - \frac{1}{2} a^{2} - \frac{7}{36} a - \frac{1}{9}$, $\frac{1}{2142720} a^{14} + \frac{433}{1071360} a^{13} - \frac{773}{714240} a^{12} + \frac{437}{1071360} a^{11} + \frac{629}{714240} a^{10} - \frac{245389}{2142720} a^{9} + \frac{139771}{714240} a^{8} - \frac{332173}{1071360} a^{7} + \frac{779659}{2142720} a^{6} + \frac{486743}{1071360} a^{5} - \frac{424387}{2142720} a^{4} + \frac{15245}{47616} a^{3} + \frac{45937}{214272} a^{2} + \frac{4381}{26784} a - \frac{269}{13392}$, $\frac{1}{3516997061593045480152404186202155235655296614400} a^{15} + \frac{661427499424378035980676777909360573579601}{3516997061593045480152404186202155235655296614400} a^{14} - \frac{1716123092812652778558142891972850421323080913}{3516997061593045480152404186202155235655296614400} a^{13} - \frac{4923393013274050224169413370148382720473266199}{3516997061593045480152404186202155235655296614400} a^{12} + \frac{170518858191348997026033236575324877441955377}{95053974637649877841956869897355546909602611200} a^{11} - \frac{187192926408503699609703767350863551327392303}{35169970615930454801524041862021552356552966144} a^{10} + \frac{157678427807777394230672689625102642214346616903}{1758498530796522740076202093101077617827648307200} a^{9} - \frac{292842150184479946164100812618599426752286098139}{3516997061593045480152404186202155235655296614400} a^{8} - \frac{167273665641763072480509338190791019487449544163}{390777451288116164461378242911350581739477401600} a^{7} - \frac{205455903828383091775349739593357266169110694063}{1172332353864348493384134728734051745218432204800} a^{6} - \frac{312520692942099490177803336913826933645515668637}{703399412318609096030480837240431047131059322880} a^{5} - \frac{53873671558195172039850636226132310716227382717}{175849853079652274007620209310107761782764830720} a^{4} + \frac{6270027766552973346162785529995423970464019971}{22690303623180938581628414104530033778421268480} a^{3} - \frac{61404229493546223271559647481477934027952866233}{351699706159304548015240418620215523565529661440} a^{2} - \frac{1947924924699360891715673052428334393886240003}{8792492653982613700381010465505388089138241536} a + \frac{145439841879812360669078364782010627738112633}{4396246326991306850190505232752694044569120768}$

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:  \( 1427890714800000 \) (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 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{6}$ R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ 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