Properties

Label 16.16.612...689.1
Degree $16$
Signature $[16, 0]$
Discriminant $6.123\times 10^{39}$
Root discriminant $306.68$
Ramified primes $13, 41$
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 - x^15 - 234*x^14 + 509*x^13 + 18337*x^12 - 57176*x^11 - 592421*x^10 + 2200433*x^9 + 8770762*x^8 - 37263645*x^7 - 58997921*x^6 + 293141240*x^5 + 166860260*x^4 - 1019696592*x^3 - 220707968*x^2 + 1184846848*x + 504881152)
 
gp: K = bnfinit(x^16 - x^15 - 234*x^14 + 509*x^13 + 18337*x^12 - 57176*x^11 - 592421*x^10 + 2200433*x^9 + 8770762*x^8 - 37263645*x^7 - 58997921*x^6 + 293141240*x^5 + 166860260*x^4 - 1019696592*x^3 - 220707968*x^2 + 1184846848*x + 504881152, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![504881152, 1184846848, -220707968, -1019696592, 166860260, 293141240, -58997921, -37263645, 8770762, 2200433, -592421, -57176, 18337, 509, -234, -1, 1]);
 

\( x^{16} - x^{15} - 234 x^{14} + 509 x^{13} + 18337 x^{12} - 57176 x^{11} - 592421 x^{10} + 2200433 x^{9} + 8770762 x^{8} - 37263645 x^{7} - 58997921 x^{6} + 293141240 x^{5} + 166860260 x^{4} - 1019696592 x^{3} - 220707968 x^{2} + 1184846848 x + 504881152 \)

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:  \(6123007382888435990757129497254763904689\)\(\medspace = 13^{14}\cdot 41^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $306.68$
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:  $13, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{5} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{368} a^{12} + \frac{1}{184} a^{11} - \frac{5}{184} a^{10} - \frac{37}{368} a^{9} + \frac{21}{184} a^{8} + \frac{1}{23} a^{7} + \frac{15}{368} a^{6} - \frac{37}{184} a^{5} + \frac{45}{184} a^{4} + \frac{53}{368} a^{3} - \frac{89}{184} a^{2} - \frac{8}{23} a - \frac{8}{23}$, $\frac{1}{736} a^{13} + \frac{1}{23} a^{11} + \frac{29}{736} a^{10} + \frac{35}{368} a^{9} + \frac{3}{92} a^{8} - \frac{17}{736} a^{7} + \frac{5}{46} a^{6} - \frac{11}{46} a^{5} - \frac{173}{736} a^{4} - \frac{27}{368} a^{3} - \frac{29}{92} a^{2} + \frac{4}{23} a + \frac{8}{23}$, $\frac{1}{29440} a^{14} + \frac{3}{5888} a^{13} - \frac{13}{14720} a^{12} - \frac{67}{29440} a^{11} - \frac{1583}{29440} a^{10} - \frac{3}{32} a^{9} - \frac{7}{256} a^{8} + \frac{5153}{29440} a^{7} + \frac{89}{2944} a^{6} - \frac{7261}{29440} a^{5} - \frac{2257}{29440} a^{4} + \frac{249}{736} a^{3} + \frac{1881}{7360} a^{2} + \frac{807}{1840} a - \frac{73}{230}$, $\frac{1}{2275287989285255765262296454746663385182248960} a^{15} - \frac{5905371745969680507632381624404916257577}{2275287989285255765262296454746663385182248960} a^{14} + \frac{323498473632388108715243149999158292934847}{1137643994642627882631148227373331692591124480} a^{13} - \frac{140197717818678018378457977422192543765079}{455057597857051153052459290949332677036449792} a^{12} - \frac{3162083001787988507626718288419413936939313}{98925564751532859359230280641159277616619520} a^{11} + \frac{2111185789522879697423207141465470648413099}{35551374832582121332223382105416615393472640} a^{10} + \frac{53675477171996107591473780456516582485591135}{455057597857051153052459290949332677036449792} a^{9} + \frac{158055230445539292447252181598284516436043833}{2275287989285255765262296454746663385182248960} a^{8} + \frac{61695683815360784805532901604244650774490097}{1137643994642627882631148227373331692591124480} a^{7} + \frac{19479913315433111599434516275634775597260179}{2275287989285255765262296454746663385182248960} a^{6} - \frac{77030828769360272516755334786677385715149445}{455057597857051153052459290949332677036449792} a^{5} + \frac{8495891876419901435944339804437427793933447}{71102749665164242664446764210833230786945280} a^{4} + \frac{283173693773875490257754339084579138721246521}{568821997321313941315574113686665846295562240} a^{3} + \frac{41299443116272964019283344536854169861687209}{142205499330328485328893528421666461573890560} a^{2} + \frac{1209340571632355507829376077903121870965677}{8887843708145530333055845526354153848368160} a + \frac{939661839462485726389901646132958020486107}{2221960927036382583263961381588538462092040}$

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:  \( 222610131226000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{16}\cdot(2\pi)^{0}\cdot 222610131226000000 \cdot 2}{2\sqrt{6123007382888435990757129497254763904689}}\approx 186.441446929653$ (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

\(\Q(\sqrt{41}) \), 4.4.11647649.1, 8.8.940041681957275729.2

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ $16$ $16$ R $16$ $16$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{16}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
41Data not computed