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Properties

Label 16.16.7968862099...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 29^{4}\cdot 181^{4}$
Root discriminant $98.59$
Ramified primes $2, 3, 5, 29, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3$ (as 16T203)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4182359056, -5263277392, -8596723528, -325075504, 2293812968, 325908724, -275403616, -44414652, 18559353, 2695196, -745869, -81548, 17428, 1162, -212, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 212*x^14 + 1162*x^13 + 17428*x^12 - 81548*x^11 - 745869*x^10 + 2695196*x^9 + 18559353*x^8 - 44414652*x^7 - 275403616*x^6 + 325908724*x^5 + 2293812968*x^4 - 325075504*x^3 - 8596723528*x^2 - 5263277392*x + 4182359056)
 
gp: K = bnfinit(x^16 - 6*x^15 - 212*x^14 + 1162*x^13 + 17428*x^12 - 81548*x^11 - 745869*x^10 + 2695196*x^9 + 18559353*x^8 - 44414652*x^7 - 275403616*x^6 + 325908724*x^5 + 2293812968*x^4 - 325075504*x^3 - 8596723528*x^2 - 5263277392*x + 4182359056, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 212 x^{14} + 1162 x^{13} + 17428 x^{12} - 81548 x^{11} - 745869 x^{10} + 2695196 x^{9} + 18559353 x^{8} - 44414652 x^{7} - 275403616 x^{6} + 325908724 x^{5} + 2293812968 x^{4} - 325075504 x^{3} - 8596723528 x^{2} - 5263277392 x + 4182359056 \)

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:  \(79688620999701608976000000000000=2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 29^{4}\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.59$
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:  $2, 3, 5, 29, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{12} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{5}{12} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{13} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{5}{12} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{20136} a^{14} + \frac{29}{839} a^{13} - \frac{22}{2517} a^{12} + \frac{41}{10068} a^{11} + \frac{245}{1678} a^{10} - \frac{521}{5034} a^{9} - \frac{653}{20136} a^{8} + \frac{4559}{10068} a^{7} - \frac{9455}{20136} a^{6} - \frac{3595}{10068} a^{5} - \frac{2239}{5034} a^{4} - \frac{1169}{5034} a^{3} - \frac{407}{5034} a^{2} - \frac{436}{2517} a + \frac{568}{2517}$, $\frac{1}{25338737260398116705956689361000265667775624105368024} a^{15} - \frac{1299886823250599566675263617602506552766939375}{103847283854090642237527415413935515031867311907246} a^{14} - \frac{521677378274693024485885558612043093162523608557273}{12669368630199058352978344680500132833887812052684012} a^{13} - \frac{389700670046291053101754866801232098850906435721449}{12669368630199058352978344680500132833887812052684012} a^{12} - \frac{32084256317881425094910741288799741475287366466117}{333404437636817325078377491592108758786521369807474} a^{11} - \frac{4308386734882250741885380983045653012054272804728}{55567406272802887513062915265351459797753561634579} a^{10} + \frac{10323094041896111797997274869356986658273030358106331}{25338737260398116705956689361000265667775624105368024} a^{9} + \frac{1175568065700557539260339071898106023463239802462513}{12669368630199058352978344680500132833887812052684012} a^{8} - \frac{1607843282165765075832508348137894435127968236682277}{25338737260398116705956689361000265667775624105368024} a^{7} - \frac{2049242692992450442688705555702221218667936434663699}{4223122876733019450992781560166710944629270684228004} a^{6} - \frac{884485111651979962313012701383090177558082294017343}{12669368630199058352978344680500132833887812052684012} a^{5} - \frac{777877224377716402216149377316496919727514816245007}{2111561438366509725496390780083355472314635342114002} a^{4} - \frac{677055905680620633524960538830821314251750029843643}{6334684315099529176489172340250066416943906026342006} a^{3} - \frac{431312275953460022137335087604727381422343175797901}{1055780719183254862748195390041677736157317671057001} a^{2} - \frac{237451297347688757001036034385272078467411675174624}{1055780719183254862748195390041677736157317671057001} a - \frac{1205275385063104953076147860193361612540177910083714}{3167342157549764588244586170125033208471953013171003}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 53162893092.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T203):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 41 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{15})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{60})^+\)

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: 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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
181Data not computed