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Properties

Label 16.16.1816151322...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{14}\cdot 29^{14}$
Root discriminant $77.84$
Ramified primes $5, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36605, -305460, 412000, 249570, -657550, 66660, 358055, -120715, -81859, 42278, 6508, -5994, 165, 351, -37, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 37*x^14 + 351*x^13 + 165*x^12 - 5994*x^11 + 6508*x^10 + 42278*x^9 - 81859*x^8 - 120715*x^7 + 358055*x^6 + 66660*x^5 - 657550*x^4 + 249570*x^3 + 412000*x^2 - 305460*x + 36605)
 
gp: K = bnfinit(x^16 - 7*x^15 - 37*x^14 + 351*x^13 + 165*x^12 - 5994*x^11 + 6508*x^10 + 42278*x^9 - 81859*x^8 - 120715*x^7 + 358055*x^6 + 66660*x^5 - 657550*x^4 + 249570*x^3 + 412000*x^2 - 305460*x + 36605, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 37 x^{14} + 351 x^{13} + 165 x^{12} - 5994 x^{11} + 6508 x^{10} + 42278 x^{9} - 81859 x^{8} - 120715 x^{7} + 358055 x^{6} + 66660 x^{5} - 657550 x^{4} + 249570 x^{3} + 412000 x^{2} - 305460 x + 36605 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1816151322484127584722900390625=5^{14}\cdot 29^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.84$
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:  $5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{109148385864707607720381839} a^{15} + \frac{7521411150707242766704244}{109148385864707607720381839} a^{14} - \frac{19589269969206395121166223}{109148385864707607720381839} a^{13} + \frac{2695407524145934335389499}{109148385864707607720381839} a^{12} + \frac{2186031860748869809834835}{109148385864707607720381839} a^{11} - \frac{17374220952561309298681603}{109148385864707607720381839} a^{10} - \frac{32007417248685643927382569}{109148385864707607720381839} a^{9} - \frac{40322734764399094086717904}{109148385864707607720381839} a^{8} + \frac{2328955922685625028055292}{109148385864707607720381839} a^{7} + \frac{13700570918340081598609710}{109148385864707607720381839} a^{6} - \frac{38476790412791430334891198}{109148385864707607720381839} a^{5} - \frac{48358053739161463333053414}{109148385864707607720381839} a^{4} + \frac{5786999524422856428143561}{109148385864707607720381839} a^{3} + \frac{45815275926451391989779231}{109148385864707607720381839} a^{2} - \frac{22966061995780035742231412}{109148385864707607720381839} a + \frac{15028937224736062642483552}{109148385864707607720381839}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $15$
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:  \( 5465061806.73 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.9294114390625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$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}$