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Properties

Label 16.16.6788684232...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{16}\cdot 5^{14}\cdot 61^{8}\cdot 97^{4}$
Root discriminant $200.44$
Ramified primes $2, 5, 61, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T852

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![157226892001, -81401551558, -225185228471, -4051510228, 65368437133, 9143434322, -5649150424, -810222424, 244580013, 27242342, -5963666, -399734, 78273, 2274, -469, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 469*x^14 + 2274*x^13 + 78273*x^12 - 399734*x^11 - 5963666*x^10 + 27242342*x^9 + 244580013*x^8 - 810222424*x^7 - 5649150424*x^6 + 9143434322*x^5 + 65368437133*x^4 - 4051510228*x^3 - 225185228471*x^2 - 81401551558*x + 157226892001)
 
gp: K = bnfinit(x^16 - 4*x^15 - 469*x^14 + 2274*x^13 + 78273*x^12 - 399734*x^11 - 5963666*x^10 + 27242342*x^9 + 244580013*x^8 - 810222424*x^7 - 5649150424*x^6 + 9143434322*x^5 + 65368437133*x^4 - 4051510228*x^3 - 225185228471*x^2 - 81401551558*x + 157226892001, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 469 x^{14} + 2274 x^{13} + 78273 x^{12} - 399734 x^{11} - 5963666 x^{10} + 27242342 x^{9} + 244580013 x^{8} - 810222424 x^{7} - 5649150424 x^{6} + 9143434322 x^{5} + 65368437133 x^{4} - 4051510228 x^{3} - 225185228471 x^{2} - 81401551558 x + 157226892001 \)

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:  \(6788684232836496753984400000000000000=2^{16}\cdot 5^{14}\cdot 61^{8}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $200.44$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{13303190835948717063749184842983900243904311705134997865424175425769796506419841} a^{15} + \frac{3292724730659759983811626447768147833571310710578530995666652610542344232455432}{13303190835948717063749184842983900243904311705134997865424175425769796506419841} a^{14} + \frac{2619870796420038731339152082482201009669104439361621401610771366110726601905463}{13303190835948717063749184842983900243904311705134997865424175425769796506419841} a^{13} + \frac{5114839270828945996793395571789568659075845426070732772483651320690398673710594}{13303190835948717063749184842983900243904311705134997865424175425769796506419841} a^{12} - \frac{1173696080519869386275502793338227576044442777765250522049058763652440814538321}{13303190835948717063749184842983900243904311705134997865424175425769796506419841} a^{11} - \frac{1722020739351950709893423036212227291439313283731375348049102363944493333426321}{13303190835948717063749184842983900243904311705134997865424175425769796506419841} a^{10} - \frac{149486076755930946750009024114890394153911028888169501389566603510797183705946}{13303190835948717063749184842983900243904311705134997865424175425769796506419841} a^{9} - \frac{5777655263236179519662900008970196518524490825373462393452110894462205057758865}{13303190835948717063749184842983900243904311705134997865424175425769796506419841} a^{8} - \frac{1201238606640576029963919621243163532251450565175624044709437707699379023626579}{13303190835948717063749184842983900243904311705134997865424175425769796506419841} a^{7} + \frac{5392812149845410556700998641523810269563221718091451514283137132111915212946765}{13303190835948717063749184842983900243904311705134997865424175425769796506419841} a^{6} + \frac{3604153399883443350009059870531917245556822469969920078651034619638456436825249}{13303190835948717063749184842983900243904311705134997865424175425769796506419841} a^{5} - \frac{5228263381581409344157520378600359423679772743433281460631212383326587664795075}{13303190835948717063749184842983900243904311705134997865424175425769796506419841} a^{4} + \frac{5223299586514975072380967472641977921094842729682552931583881516154449759100308}{13303190835948717063749184842983900243904311705134997865424175425769796506419841} a^{3} - \frac{4774798144932032232579627236749516040061812572710998619099795858000564167765814}{13303190835948717063749184842983900243904311705134997865424175425769796506419841} a^{2} - \frac{714449435022649170052720126323384090964230859960463309155143264226487227645042}{13303190835948717063749184842983900243904311705134997865424175425769796506419841} a - \frac{516092956638850238440912251446335662606650875633340604296107235788956675809626}{13303190835948717063749184842983900243904311705134997865424175425769796506419841}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 8472302079270 \) (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}$