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Properties

Label 16.0.24722989956...4209.6
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 59^{8}$
Root discriminant $91.64$
Ramified primes $17, 59$
Class number $217848$ (GRH)
Class group $[217848]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5933446319, -1664461289, 2517198244, -622686499, 486803774, -104874598, 55839765, -10300950, 4145112, -637603, 203540, -24922, 6445, -572, 120, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 120*x^14 - 572*x^13 + 6445*x^12 - 24922*x^11 + 203540*x^10 - 637603*x^9 + 4145112*x^8 - 10300950*x^7 + 55839765*x^6 - 104874598*x^5 + 486803774*x^4 - 622686499*x^3 + 2517198244*x^2 - 1664461289*x + 5933446319)
 
gp: K = bnfinit(x^16 - 6*x^15 + 120*x^14 - 572*x^13 + 6445*x^12 - 24922*x^11 + 203540*x^10 - 637603*x^9 + 4145112*x^8 - 10300950*x^7 + 55839765*x^6 - 104874598*x^5 + 486803774*x^4 - 622686499*x^3 + 2517198244*x^2 - 1664461289*x + 5933446319, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 120 x^{14} - 572 x^{13} + 6445 x^{12} - 24922 x^{11} + 203540 x^{10} - 637603 x^{9} + 4145112 x^{8} - 10300950 x^{7} + 55839765 x^{6} - 104874598 x^{5} + 486803774 x^{4} - 622686499 x^{3} + 2517198244 x^{2} - 1664461289 x + 5933446319 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(24722989956581301387479701814209=17^{14}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.64$
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:  $17, 59$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1003=17\cdot 59\)
Dirichlet character group:    $\lbrace$$\chi_{1003}(1,·)$, $\chi_{1003}(648,·)$, $\chi_{1003}(650,·)$, $\chi_{1003}(591,·)$, $\chi_{1003}(412,·)$, $\chi_{1003}(353,·)$, $\chi_{1003}(355,·)$, $\chi_{1003}(1002,·)$, $\chi_{1003}(237,·)$, $\chi_{1003}(943,·)$, $\chi_{1003}(178,·)$, $\chi_{1003}(117,·)$, $\chi_{1003}(886,·)$, $\chi_{1003}(825,·)$, $\chi_{1003}(60,·)$, $\chi_{1003}(766,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{31668328779672326831155258657576320261155382047746} a^{15} - \frac{2194721697154506650484092819917534399779706269764}{15834164389836163415577629328788160130577691023873} a^{14} - \frac{1060455543552955148667535473546727325981693974254}{15834164389836163415577629328788160130577691023873} a^{13} - \frac{1053120036848987616542650363544919987580716171857}{15834164389836163415577629328788160130577691023873} a^{12} - \frac{3249841053066214365511883117661532576384125927848}{15834164389836163415577629328788160130577691023873} a^{11} - \frac{7893079354275629697038504331968227728617214155713}{15834164389836163415577629328788160130577691023873} a^{10} + \frac{5646125056942933913557452272897981499097113092301}{15834164389836163415577629328788160130577691023873} a^{9} + \frac{360943249823633799999379801192119770867906958137}{15834164389836163415577629328788160130577691023873} a^{8} - \frac{5729304487124826912806735536645024540626191947889}{15834164389836163415577629328788160130577691023873} a^{7} - \frac{6762820963646789922211054476432687742539603790527}{15834164389836163415577629328788160130577691023873} a^{6} - \frac{4402372083811010020624631847014958292046157384271}{15834164389836163415577629328788160130577691023873} a^{5} - \frac{221430102703928011530055519043248982579086773306}{15834164389836163415577629328788160130577691023873} a^{4} - \frac{7366229829725078977788389388575172737589062506286}{15834164389836163415577629328788160130577691023873} a^{3} - \frac{5655779634340098624409147946523674438840410956768}{15834164389836163415577629328788160130577691023873} a^{2} + \frac{3374483266213481257863744209320779910659623459185}{15834164389836163415577629328788160130577691023873} a + \frac{7025930302678194030462246300332023905694546090417}{31668328779672326831155258657576320261155382047746}$

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

Class group and class number

$C_{217848}$, which has order $217848$ (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:  $7$
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:  \( 3640.01221338 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-59}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-1003}) \), \(\Q(\sqrt{17}, \sqrt{-59})\), 4.4.4913.1, 4.0.17102153.2, 8.0.292483637235409.3, 8.0.4972221833001953.2, \(\Q(\zeta_{17})^+\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
$59$59.8.4.1$x^{8} + 97468 x^{4} - 205379 x^{2} + 2375002756$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
59.8.4.1$x^{8} + 97468 x^{4} - 205379 x^{2} + 2375002756$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$