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Properties

Label 16.0.15085984382...0000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 5^{12}\cdot 37^{8}$
Root discriminant $136.82$
Ramified primes $2, 5, 37$
Class number $1206400$ (GRH)
Class group $[40, 30160]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![85468811201, -22976725744, 28576785148, -6503562836, 4239589780, -793737240, 359963494, -53702132, 18981756, -2150452, 642284, -51308, 13793, -684, 174, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 174*x^14 - 684*x^13 + 13793*x^12 - 51308*x^11 + 642284*x^10 - 2150452*x^9 + 18981756*x^8 - 53702132*x^7 + 359963494*x^6 - 793737240*x^5 + 4239589780*x^4 - 6503562836*x^3 + 28576785148*x^2 - 22976725744*x + 85468811201)
 
gp: K = bnfinit(x^16 - 4*x^15 + 174*x^14 - 684*x^13 + 13793*x^12 - 51308*x^11 + 642284*x^10 - 2150452*x^9 + 18981756*x^8 - 53702132*x^7 + 359963494*x^6 - 793737240*x^5 + 4239589780*x^4 - 6503562836*x^3 + 28576785148*x^2 - 22976725744*x + 85468811201, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 174 x^{14} - 684 x^{13} + 13793 x^{12} - 51308 x^{11} + 642284 x^{10} - 2150452 x^{9} + 18981756 x^{8} - 53702132 x^{7} + 359963494 x^{6} - 793737240 x^{5} + 4239589780 x^{4} - 6503562836 x^{3} + 28576785148 x^{2} - 22976725744 x + 85468811201 \)

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:  \(15085984382462633967616000000000000=2^{44}\cdot 5^{12}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $136.82$
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:  $2, 5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2960=2^{4}\cdot 5\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{2960}(1,·)$, $\chi_{2960}(517,·)$, $\chi_{2960}(2369,·)$, $\chi_{2960}(1481,·)$, $\chi_{2960}(1997,·)$, $\chi_{2960}(813,·)$, $\chi_{2960}(1553,·)$, $\chi_{2960}(149,·)$, $\chi_{2960}(1629,·)$, $\chi_{2960}(741,·)$, $\chi_{2960}(1257,·)$, $\chi_{2960}(2221,·)$, $\chi_{2960}(2737,·)$, $\chi_{2960}(2293,·)$, $\chi_{2960}(73,·)$, $\chi_{2960}(889,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{109} a^{12} - \frac{40}{109} a^{11} + \frac{42}{109} a^{10} - \frac{39}{109} a^{9} + \frac{14}{109} a^{8} - \frac{21}{109} a^{7} + \frac{11}{109} a^{6} + \frac{36}{109} a^{5} - \frac{43}{109} a^{4} - \frac{11}{109} a^{3} + \frac{25}{109} a^{2} - \frac{31}{109} a + \frac{4}{109}$, $\frac{1}{109} a^{13} - \frac{32}{109} a^{11} + \frac{6}{109} a^{10} - \frac{20}{109} a^{9} - \frac{6}{109} a^{8} + \frac{43}{109} a^{7} + \frac{40}{109} a^{6} - \frac{20}{109} a^{5} + \frac{13}{109} a^{4} + \frac{21}{109} a^{3} - \frac{12}{109} a^{2} - \frac{37}{109} a + \frac{51}{109}$, $\frac{1}{8849875142340094795743} a^{14} + \frac{632843817973878529}{8849875142340094795743} a^{13} + \frac{6328566686942560738}{2949958380780031598581} a^{12} + \frac{2523962178623335112950}{8849875142340094795743} a^{11} - \frac{1762966104116163645653}{8849875142340094795743} a^{10} + \frac{1702713026616022116028}{8849875142340094795743} a^{9} + \frac{936817142190997290353}{8849875142340094795743} a^{8} + \frac{2860643667466374107797}{8849875142340094795743} a^{7} + \frac{775125466050583717444}{8849875142340094795743} a^{6} + \frac{691843030816447371655}{8849875142340094795743} a^{5} + \frac{115094850995300837908}{8849875142340094795743} a^{4} + \frac{763242086095990748165}{2949958380780031598581} a^{3} - \frac{2039099366401787932105}{8849875142340094795743} a^{2} + \frac{3660495167149688554352}{8849875142340094795743} a + \frac{3439903806879883100770}{8849875142340094795743}$, $\frac{1}{115464584405688417283998440166778769437023} a^{15} - \frac{4779974396740169408}{115464584405688417283998440166778769437023} a^{14} - \frac{31764648804416296398870168436533211604}{38488194801896139094666146722259589812341} a^{13} - \frac{218373486166668385421503671792055892618}{115464584405688417283998440166778769437023} a^{12} + \frac{37459351967355377640470729614462355669932}{115464584405688417283998440166778769437023} a^{11} + \frac{21702304503444449242028952280084892394025}{115464584405688417283998440166778769437023} a^{10} + \frac{19956397295666936858301885592734030900035}{115464584405688417283998440166778769437023} a^{9} - \frac{12188214298692212928441073702634221132397}{115464584405688417283998440166778769437023} a^{8} + \frac{25645981983966218576546003893849638631981}{115464584405688417283998440166778769437023} a^{7} - \frac{56625583331952827749664133697435776138365}{115464584405688417283998440166778769437023} a^{6} - \frac{3323644926696004299369638896473198676934}{115464584405688417283998440166778769437023} a^{5} - \frac{1561519138669336032405547479923511073058}{38488194801896139094666146722259589812341} a^{4} + \frac{37656701494066982975236232721513883236833}{115464584405688417283998440166778769437023} a^{3} + \frac{42584049769100025284364307381247318890736}{115464584405688417283998440166778769437023} a^{2} - \frac{9343218194217163339652455444589783768474}{115464584405688417283998440166778769437023} a - \frac{16744142033045068994885525491870137552696}{38488194801896139094666146722259589812341}$

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

Class group and class number

$C_{40}\times C_{30160}$, which has order $1206400$ (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:  \( 12198.951274811623 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.51200.1, 4.0.171125.1, 4.0.10952000.2, 4.0.350464000.4, 4.0.350464000.2, 8.8.2621440000.1, 8.0.119946304000000.36, 8.0.122825015296000000.7

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
$37$37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$