Properties

Label 16.8.488...321.1
Degree $16$
Signature $[8, 4]$
Discriminant $4.881\times 10^{36}$
Root discriminant $196.35$
Ramified primes $11, 41$
Class number $1$ (GRH)
Class group trivial (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 - 2*x^15 - 116*x^14 + 177*x^13 + 7745*x^12 - 10421*x^11 - 424462*x^10 - 657538*x^9 + 8943832*x^8 + 23224335*x^7 - 106230562*x^6 - 311941107*x^5 + 640011076*x^4 + 1607965375*x^3 - 1656703698*x^2 - 2425324435*x + 1779444071)
 
gp: K = bnfinit(x^16 - 2*x^15 - 116*x^14 + 177*x^13 + 7745*x^12 - 10421*x^11 - 424462*x^10 - 657538*x^9 + 8943832*x^8 + 23224335*x^7 - 106230562*x^6 - 311941107*x^5 + 640011076*x^4 + 1607965375*x^3 - 1656703698*x^2 - 2425324435*x + 1779444071, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1779444071, -2425324435, -1656703698, 1607965375, 640011076, -311941107, -106230562, 23224335, 8943832, -657538, -424462, -10421, 7745, 177, -116, -2, 1]);
 

\( x^{16} - 2 x^{15} - 116 x^{14} + 177 x^{13} + 7745 x^{12} - 10421 x^{11} - 424462 x^{10} - 657538 x^{9} + 8943832 x^{8} + 23224335 x^{7} - 106230562 x^{6} - 311941107 x^{5} + 640011076 x^{4} + 1607965375 x^{3} - 1656703698 x^{2} - 2425324435 x + 1779444071 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(4880564680360454664521443587092657321\)\(\medspace = 11^{12}\cdot 41^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $196.35$
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:  $11, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{497780381670378618167911898911587624894480524954740209099692235097563929} a^{15} - \frac{121824010434318920037722971921310557775052812026440774165696600110323356}{497780381670378618167911898911587624894480524954740209099692235097563929} a^{14} - \frac{225296090067364423173907297573486028355077282919017598911138647378040266}{497780381670378618167911898911587624894480524954740209099692235097563929} a^{13} - \frac{58348660270127396672969782515064667109106456817481444475075185645853250}{497780381670378618167911898911587624894480524954740209099692235097563929} a^{12} - \frac{244094102105490103992974766011080133395917652559091354104537625197954554}{497780381670378618167911898911587624894480524954740209099692235097563929} a^{11} - \frac{224595388462840495708066978278864990969352082987015546867247049376968632}{497780381670378618167911898911587624894480524954740209099692235097563929} a^{10} - \frac{66331507242909737193514588867812857969448268423062399908254593537102690}{497780381670378618167911898911587624894480524954740209099692235097563929} a^{9} + \frac{14863695820506267347367199227429655001675497385971300670293280794654987}{497780381670378618167911898911587624894480524954740209099692235097563929} a^{8} + \frac{163905376613102636828578881800263244913291379915522045757693601657956049}{497780381670378618167911898911587624894480524954740209099692235097563929} a^{7} - \frac{172699433408266011010189545185293628023771173696371558001816222343603317}{497780381670378618167911898911587624894480524954740209099692235097563929} a^{6} - \frac{122431639060927341017754185268803650485122765914920703322360338827290028}{497780381670378618167911898911587624894480524954740209099692235097563929} a^{5} + \frac{72037335479845838059582569047186991876615008389234489776644724026821613}{497780381670378618167911898911587624894480524954740209099692235097563929} a^{4} - \frac{140780125444533059895712900632421313255390056034056677896327275461396283}{497780381670378618167911898911587624894480524954740209099692235097563929} a^{3} - \frac{52657384748378036476207476016154544515238761626776013394411732170575612}{497780381670378618167911898911587624894480524954740209099692235097563929} a^{2} - \frac{193830127596217944597608649682990157189893759493387935078587460405000653}{497780381670378618167911898911587624894480524954740209099692235097563929} a + \frac{24408970402522528586702160455251280282844272516940374259894168906135}{228025827608968675294508428269165196928300744367723412322351000960863}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 2773762487010 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 2773762487010 \cdot 1}{2\sqrt{4880564680360454664521443587092657321}}\approx 0.250474591564688$ (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

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.2851397323891721.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ R $16$ $16$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
11Data not computed
41Data not computed