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Properties

Label 16.16.1824424679...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{16}\cdot 5^{14}\cdot 61^{6}\cdot 97^{4}$
Root discriminant $119.90$
Ramified primes $2, 5, 61, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T852

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-248412799, 487879612, 82576530, -511487750, 129284605, 136830436, -47466883, -15713660, 6121920, 908910, -373958, -27226, 11610, 390, -175, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 175*x^14 + 390*x^13 + 11610*x^12 - 27226*x^11 - 373958*x^10 + 908910*x^9 + 6121920*x^8 - 15713660*x^7 - 47466883*x^6 + 136830436*x^5 + 129284605*x^4 - 511487750*x^3 + 82576530*x^2 + 487879612*x - 248412799)
 
gp: K = bnfinit(x^16 - 2*x^15 - 175*x^14 + 390*x^13 + 11610*x^12 - 27226*x^11 - 373958*x^10 + 908910*x^9 + 6121920*x^8 - 15713660*x^7 - 47466883*x^6 + 136830436*x^5 + 129284605*x^4 - 511487750*x^3 + 82576530*x^2 + 487879612*x - 248412799, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 175 x^{14} + 390 x^{13} + 11610 x^{12} - 27226 x^{11} - 373958 x^{10} + 908910 x^{9} + 6121920 x^{8} - 15713660 x^{7} - 47466883 x^{6} + 136830436 x^{5} + 129284605 x^{4} - 511487750 x^{3} + 82576530 x^{2} + 487879612 x - 248412799 \)

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:  \(1824424679612065776400000000000000=2^{16}\cdot 5^{14}\cdot 61^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $119.90$
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:  $2, 5, 61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{5} - \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{6} - \frac{1}{5} a$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{7} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{24341025351390055866503828006634668808472171687505525} a^{15} - \frac{41355006704685686992356647987879435419172487311026}{973641014055602234660153120265386752338886867500221} a^{14} + \frac{352366099493815891327617510352783028019049115376219}{4868205070278011173300765601326933761694434337501105} a^{13} - \frac{245555512858830834147701958233076612662152150493832}{4868205070278011173300765601326933761694434337501105} a^{12} + \frac{283817769432588010281339899322474355297115283800716}{4868205070278011173300765601326933761694434337501105} a^{11} + \frac{1744506570660420704153934853000674769033035642795924}{24341025351390055866503828006634668808472171687505525} a^{10} + \frac{319419286629697860546065858058813200091412700962949}{4868205070278011173300765601326933761694434337501105} a^{9} + \frac{422675278599853227569305120132780943435268605251967}{4868205070278011173300765601326933761694434337501105} a^{8} - \frac{434861167866406753234903500446976857520558015594602}{973641014055602234660153120265386752338886867500221} a^{7} + \frac{204361496185818633899350938544043801520086593979239}{4868205070278011173300765601326933761694434337501105} a^{6} - \frac{4018193262156342877681166733571460027610437946971968}{24341025351390055866503828006634668808472171687505525} a^{5} - \frac{995932163392615087214200697699125719297709873075828}{4868205070278011173300765601326933761694434337501105} a^{4} + \frac{1595732570516461520680209571606172602185553361496}{973641014055602234660153120265386752338886867500221} a^{3} - \frac{1866617078376083851778410127200311992313902288421837}{4868205070278011173300765601326933761694434337501105} a^{2} + \frac{2302015452307003335389271218146189844923964788100638}{4868205070278011173300765601326933761694434337501105} a - \frac{4643170262523610999570218212078673617713196400210943}{24341025351390055866503828006634668808472171687505525}$

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:  \( 246678315227 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T852:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n852
Character table for t16n852 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.14884000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
5Data not computed
61Data not computed
$97$97.8.4.2$x^{8} - 912673 x^{2} + 2036173463$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$