Properties

Label 16.0.350...673.1
Degree $16$
Signature $[0, 8]$
Discriminant $3.508\times 10^{43}$
Root discriminant $526.70$
Ramified primes $13, 73$
Class number $1116900$ (GRH)
Class group $[15, 74460]$ (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 80*x^14 - 285*x^13 - 278*x^12 - 2577*x^11 + 390922*x^10 - 1082129*x^9 + 9064381*x^8 - 99454552*x^7 - 680012525*x^6 - 6407302484*x^5 - 14729219316*x^4 + 64445226783*x^3 + 1579747491060*x^2 + 7868587088456*x + 16658388554993)
 
gp: K = bnfinit(x^16 - 7*x^15 + 80*x^14 - 285*x^13 - 278*x^12 - 2577*x^11 + 390922*x^10 - 1082129*x^9 + 9064381*x^8 - 99454552*x^7 - 680012525*x^6 - 6407302484*x^5 - 14729219316*x^4 + 64445226783*x^3 + 1579747491060*x^2 + 7868587088456*x + 16658388554993, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16658388554993, 7868587088456, 1579747491060, 64445226783, -14729219316, -6407302484, -680012525, -99454552, 9064381, -1082129, 390922, -2577, -278, -285, 80, -7, 1]);
 

\( x^{16} - 7 x^{15} + 80 x^{14} - 285 x^{13} - 278 x^{12} - 2577 x^{11} + 390922 x^{10} - 1082129 x^{9} + 9064381 x^{8} - 99454552 x^{7} - 680012525 x^{6} - 6407302484 x^{5} - 14729219316 x^{4} + 64445226783 x^{3} + 1579747491060 x^{2} + 7868587088456 x + 16658388554993 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(35079212407887857597919870221838606532086673\)\(\medspace = 13^{14}\cdot 73^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $526.70$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $13, 73$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{868448371600397398913820899750458491188310203645727567821722340276999529716725397453736101} a^{15} - \frac{382061391836111089449840259312762528112605466620151376852758128468553439130756046788680337}{868448371600397398913820899750458491188310203645727567821722340276999529716725397453736101} a^{14} - \frac{230465825699783818287377261283240744094355180976452672507923236253222041193098310452699524}{868448371600397398913820899750458491188310203645727567821722340276999529716725397453736101} a^{13} - \frac{411122247071003244110343120618322177755639330671396351901616640259215661099953378025001184}{868448371600397398913820899750458491188310203645727567821722340276999529716725397453736101} a^{12} + \frac{226270958912684777370838393930125454499477538489438964847897476061454646637644491081990263}{868448371600397398913820899750458491188310203645727567821722340276999529716725397453736101} a^{11} + \frac{259067941962452718147540952037428335250371997606099961259624431114275815620928923602712798}{868448371600397398913820899750458491188310203645727567821722340276999529716725397453736101} a^{10} + \frac{107773287392125798104701856483699270967825455755508288298315120973053447536825446987814409}{868448371600397398913820899750458491188310203645727567821722340276999529716725397453736101} a^{9} - \frac{138400912582272761473072538898687134547104193690052204912753998956006702130418100783727118}{868448371600397398913820899750458491188310203645727567821722340276999529716725397453736101} a^{8} - \frac{197237564687668891290674857569566229667579642676763668141622393822108507700351815719200469}{868448371600397398913820899750458491188310203645727567821722340276999529716725397453736101} a^{7} - \frac{278635675582024976786411766580954916122636513674431946954461336790157791488854259752671822}{868448371600397398913820899750458491188310203645727567821722340276999529716725397453736101} a^{6} + \frac{199692469311235131133067481252891471519780373848559972923753369389149618509573741168374876}{868448371600397398913820899750458491188310203645727567821722340276999529716725397453736101} a^{5} + \frac{144714336318212187238701718362184845542450337330574581846427408855564250328738269499699345}{868448371600397398913820899750458491188310203645727567821722340276999529716725397453736101} a^{4} + \frac{254660980262530405504774766282347049186045960859190779712356513384892853801012236955157790}{868448371600397398913820899750458491188310203645727567821722340276999529716725397453736101} a^{3} + \frac{108865172540882542262851396314338517011829454600762013263554789054974728503415631359623366}{868448371600397398913820899750458491188310203645727567821722340276999529716725397453736101} a^{2} - \frac{18013051769959533702845845315182764884429774657575066985387148880415810810381578171913416}{868448371600397398913820899750458491188310203645727567821722340276999529716725397453736101} a + \frac{107387521018690343767171679498531516023984032818653302996166151917002959403458834311273930}{868448371600397398913820899750458491188310203645727567821722340276999529716725397453736101}$

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

Class group and class number

$C_{15}\times C_{74460}$, which has order $1116900$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 487222110.331 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 487222110.331 \cdot 1116900}{2\sqrt{35079212407887857597919870221838606532086673}}\approx 0.111590009258$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.65743873.1, 8.8.53323682598564071473.2

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ $16$ R $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
13Data not computed
73Data not computed