Properties

Label 16.16.543...625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5.438\times 10^{37}$
Root discriminant $228.28$
Ramified primes $5, 73$
Class number $2$ (GRH)
Class group $[2]$ (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 - 3*x^15 - 249*x^14 + 1468*x^13 + 14951*x^12 - 127824*x^11 - 146116*x^10 + 3524299*x^9 - 6618428*x^8 - 25092711*x^7 + 111980624*x^6 - 122939949*x^5 - 71984014*x^4 + 233617078*x^3 - 109771874*x^2 - 52840783*x + 40491571)
 
gp: K = bnfinit(x^16 - 3*x^15 - 249*x^14 + 1468*x^13 + 14951*x^12 - 127824*x^11 - 146116*x^10 + 3524299*x^9 - 6618428*x^8 - 25092711*x^7 + 111980624*x^6 - 122939949*x^5 - 71984014*x^4 + 233617078*x^3 - 109771874*x^2 - 52840783*x + 40491571, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![40491571, -52840783, -109771874, 233617078, -71984014, -122939949, 111980624, -25092711, -6618428, 3524299, -146116, -127824, 14951, 1468, -249, -3, 1]);
 

\( x^{16} - 3 x^{15} - 249 x^{14} + 1468 x^{13} + 14951 x^{12} - 127824 x^{11} - 146116 x^{10} + 3524299 x^{9} - 6618428 x^{8} - 25092711 x^{7} + 111980624 x^{6} - 122939949 x^{5} - 71984014 x^{4} + 233617078 x^{3} - 109771874 x^{2} - 52840783 x + 40491571 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[16, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(54377966460580450275755400738525390625\)\(\medspace = 5^{14}\cdot 73^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $228.28$
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:  $5, 73$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{5} + \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{15} a^{12} - \frac{1}{15} a^{11} - \frac{1}{15} a^{10} - \frac{1}{15} a^{9} - \frac{1}{15} a^{8} + \frac{4}{15} a^{7} + \frac{7}{15} a^{6} + \frac{2}{15} a^{5} + \frac{2}{5} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{7}{15} a - \frac{1}{15}$, $\frac{1}{15} a^{13} + \frac{1}{15} a^{11} + \frac{1}{15} a^{10} + \frac{1}{15} a^{9} + \frac{1}{3} a^{7} + \frac{1}{5} a^{6} - \frac{7}{15} a^{5} + \frac{4}{15} a^{4} - \frac{4}{15} a^{3} + \frac{2}{5} a^{2} + \frac{7}{15} a - \frac{4}{15}$, $\frac{1}{1635} a^{14} + \frac{2}{327} a^{13} - \frac{5}{327} a^{12} + \frac{121}{1635} a^{11} + \frac{4}{1635} a^{10} + \frac{41}{545} a^{9} - \frac{152}{1635} a^{8} + \frac{253}{545} a^{7} + \frac{43}{109} a^{6} - \frac{532}{1635} a^{5} - \frac{74}{545} a^{4} + \frac{47}{545} a^{3} + \frac{389}{1635} a^{2} - \frac{661}{1635} a + \frac{133}{1635}$, $\frac{1}{261236987487992030198280695159045794111186305} a^{15} + \frac{22087419264295624135445885849402609769452}{261236987487992030198280695159045794111186305} a^{14} + \frac{1039307852952930647146600399034563930051826}{87078995829330676732760231719681931370395435} a^{13} + \frac{4006211087324096671968264801529177554847436}{261236987487992030198280695159045794111186305} a^{12} - \frac{5941671373268854164683085214892528197889317}{87078995829330676732760231719681931370395435} a^{11} - \frac{2654790616160829418190588660642169059190064}{261236987487992030198280695159045794111186305} a^{10} + \frac{19211103241354389688468919031586893209020649}{261236987487992030198280695159045794111186305} a^{9} - \frac{928434962376635664358096803555315594357935}{52247397497598406039656139031809158822237261} a^{8} + \frac{6111984390391135563190430098795349660037551}{261236987487992030198280695159045794111186305} a^{7} - \frac{10991190189158790916274457983758354773899068}{52247397497598406039656139031809158822237261} a^{6} + \frac{2793741270065420390538032303564401734912392}{87078995829330676732760231719681931370395435} a^{5} - \frac{6003488684257877991985038523932776011957817}{87078995829330676732760231719681931370395435} a^{4} - \frac{123158590282372844488566075376979933580321614}{261236987487992030198280695159045794111186305} a^{3} - \frac{9270834658936739265957063717490398573843666}{87078995829330676732760231719681931370395435} a^{2} + \frac{97959159891088258686708426115417747416385852}{261236987487992030198280695159045794111186305} a + \frac{261671342438709537347955970539812292699364}{1040784810709131594415460936888628661797555}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $15$
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:  \( 17978470399500 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{16}\cdot(2\pi)^{0}\cdot 17978470399500 \cdot 2}{2\sqrt{54377966460580450275755400738525390625}}\approx 0.159779548184107$ (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

\(\Q(\sqrt{73}) \), 4.4.9725425.1, 8.8.172615601860890625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$

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
5Data not computed
73Data not computed