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Properties

Label 16.0.60554479642...9984.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 11^{8}\cdot 17^{14}$
Root discriminant $111.91$
Ramified primes $2, 11, 17$
Class number $1008000$ (GRH)
Class group $[5, 120, 1680]$ (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![104203730401, -8015671568, 30257685388, -2066649212, 4001369686, -238996272, 314665218, -16070158, 16099839, -679232, 549386, -18086, 12233, -282, 163, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 163*x^14 - 282*x^13 + 12233*x^12 - 18086*x^11 + 549386*x^10 - 679232*x^9 + 16099839*x^8 - 16070158*x^7 + 314665218*x^6 - 238996272*x^5 + 4001369686*x^4 - 2066649212*x^3 + 30257685388*x^2 - 8015671568*x + 104203730401)
 
gp: K = bnfinit(x^16 - 2*x^15 + 163*x^14 - 282*x^13 + 12233*x^12 - 18086*x^11 + 549386*x^10 - 679232*x^9 + 16099839*x^8 - 16070158*x^7 + 314665218*x^6 - 238996272*x^5 + 4001369686*x^4 - 2066649212*x^3 + 30257685388*x^2 - 8015671568*x + 104203730401, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 163 x^{14} - 282 x^{13} + 12233 x^{12} - 18086 x^{11} + 549386 x^{10} - 679232 x^{9} + 16099839 x^{8} - 16070158 x^{7} + 314665218 x^{6} - 238996272 x^{5} + 4001369686 x^{4} - 2066649212 x^{3} + 30257685388 x^{2} - 8015671568 x + 104203730401 \)

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:  \(605544796424780341033364025769984=2^{24}\cdot 11^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $111.91$
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:  $2, 11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1496=2^{3}\cdot 11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1496}(1,·)$, $\chi_{1496}(1409,·)$, $\chi_{1496}(461,·)$, $\chi_{1496}(1165,·)$, $\chi_{1496}(1233,·)$, $\chi_{1496}(21,·)$, $\chi_{1496}(89,·)$, $\chi_{1496}(285,·)$, $\chi_{1496}(353,·)$, $\chi_{1496}(529,·)$, $\chi_{1496}(637,·)$, $\chi_{1496}(373,·)$, $\chi_{1496}(441,·)$, $\chi_{1496}(705,·)$, $\chi_{1496}(1341,·)$, $\chi_{1496}(1429,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{10155719139219844039848732285070377198178227955933204769} a^{15} - \frac{4234238032704219855270882707038796527479375597961203099}{10155719139219844039848732285070377198178227955933204769} a^{14} - \frac{3909652011699257101298196022865666518052411230418259338}{10155719139219844039848732285070377198178227955933204769} a^{13} - \frac{3834585560588777401619314134226301893530675474016296288}{10155719139219844039848732285070377198178227955933204769} a^{12} + \frac{4002294370329170333551575730514104475415835973901105341}{10155719139219844039848732285070377198178227955933204769} a^{11} + \frac{1713112009227070804142222558163097107520234554286749605}{10155719139219844039848732285070377198178227955933204769} a^{10} - \frac{3935740832054336091406356485952606614750538076470616530}{10155719139219844039848732285070377198178227955933204769} a^{9} + \frac{4215049308926653563060646340606102918692679063723664783}{10155719139219844039848732285070377198178227955933204769} a^{8} - \frac{1976561231545632675875629681678425751820068378906548569}{10155719139219844039848732285070377198178227955933204769} a^{7} - \frac{875449942396107393179197727296628465239058816948189757}{10155719139219844039848732285070377198178227955933204769} a^{6} + \frac{2213941865697759580967233186684213886061535074271892035}{10155719139219844039848732285070377198178227955933204769} a^{5} + \frac{3634507347192137355491160669858587740781013280591858525}{10155719139219844039848732285070377198178227955933204769} a^{4} + \frac{3696718339261413975170396063420412498418765338838104097}{10155719139219844039848732285070377198178227955933204769} a^{3} - \frac{4032346632321995058345828145062245480020827025738037535}{10155719139219844039848732285070377198178227955933204769} a^{2} - \frac{3042180521031791725740998432237708937419926534103774872}{10155719139219844039848732285070377198178227955933204769} a - \frac{3592111591578118584388441503483329236798003463839726842}{10155719139219844039848732285070377198178227955933204769}$

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

Class group and class number

$C_{5}\times C_{120}\times C_{1680}$, which has order $1008000$ (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:  \( 3640.012213375973 \) (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

\(\Q(\sqrt{-374}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{17}, \sqrt{-22})\), 4.4.4913.1, 4.0.38046272.6, 8.0.1447518813097984.43, 8.0.24607819822665728.4, \(\Q(\zeta_{17})^+\)

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
11Data not computed
17Data not computed