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Properties

Label 16.0.47991168245...000.26
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{12}\cdot 17^{8}$
Root discriminant $95.52$
Ramified primes $2, 3, 5, 17$
Class number $117504$ (GRH)
Class group $[2, 6, 12, 816]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![24022900, 26637000, 26809700, 15805800, 8159670, 2631000, 700670, 49800, -2019, -2100, 4442, 1440, 284, -60, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 - 60*x^13 + 284*x^12 + 1440*x^11 + 4442*x^10 - 2100*x^9 - 2019*x^8 + 49800*x^7 + 700670*x^6 + 2631000*x^5 + 8159670*x^4 + 15805800*x^3 + 26809700*x^2 + 26637000*x + 24022900)
 
gp: K = bnfinit(x^16 - 12*x^14 - 60*x^13 + 284*x^12 + 1440*x^11 + 4442*x^10 - 2100*x^9 - 2019*x^8 + 49800*x^7 + 700670*x^6 + 2631000*x^5 + 8159670*x^4 + 15805800*x^3 + 26809700*x^2 + 26637000*x + 24022900, 1)
 

Normalized defining polynomial

\( x^{16} - 12 x^{14} - 60 x^{13} + 284 x^{12} + 1440 x^{11} + 4442 x^{10} - 2100 x^{9} - 2019 x^{8} + 49800 x^{7} + 700670 x^{6} + 2631000 x^{5} + 8159670 x^{4} + 15805800 x^{3} + 26809700 x^{2} + 26637000 x + 24022900 \)

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:  \(47991168245852798976000000000000=2^{32}\cdot 3^{8}\cdot 5^{12}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.52$
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:  $2, 3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2040=2^{3}\cdot 3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{2040}(1,·)$, $\chi_{2040}(1157,·)$, $\chi_{2040}(203,·)$, $\chi_{2040}(1871,·)$, $\chi_{2040}(1427,·)$, $\chi_{2040}(409,·)$, $\chi_{2040}(271,·)$, $\chi_{2040}(1121,·)$, $\chi_{2040}(1123,·)$, $\chi_{2040}(679,·)$, $\chi_{2040}(239,·)$, $\chi_{2040}(307,·)$, $\chi_{2040}(1973,·)$, $\chi_{2040}(1529,·)$, $\chi_{2040}(1597,·)$, $\chi_{2040}(373,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{22} a^{10} + \frac{5}{22} a^{9} + \frac{3}{22} a^{8} + \frac{2}{11} a^{7} + \frac{9}{22} a^{6} + \frac{1}{22} a^{5} + \frac{5}{22} a^{4} - \frac{4}{11} a^{3} + \frac{2}{11} a^{2} - \frac{1}{11} a$, $\frac{1}{22} a^{11} - \frac{1}{2} a^{7} + \frac{5}{11} a$, $\frac{1}{110} a^{12} - \frac{1}{55} a^{10} - \frac{1}{11} a^{9} - \frac{3}{55} a^{8} - \frac{3}{11} a^{7} - \frac{9}{55} a^{6} + \frac{2}{11} a^{5} + \frac{1}{110} a^{4} - \frac{5}{11} a^{3} - \frac{2}{11} a^{2} - \frac{4}{11} a$, $\frac{1}{110} a^{13} - \frac{1}{55} a^{11} - \frac{1}{10} a^{9} + \frac{1}{5} a^{7} - \frac{2}{5} a^{5} + \frac{1}{11} a^{3} - \frac{2}{11} a$, $\frac{1}{43619133910} a^{14} + \frac{13255287}{4361913391} a^{13} + \frac{10151131}{43619133910} a^{12} - \frac{6446107}{8723826782} a^{11} - \frac{36308923}{2295743890} a^{10} - \frac{27071320}{229574389} a^{9} + \frac{453234679}{43619133910} a^{8} + \frac{226446359}{8723826782} a^{7} - \frac{4806352579}{21809566955} a^{6} + \frac{2106543081}{4361913391} a^{5} - \frac{6598908131}{21809566955} a^{4} + \frac{1358453363}{4361913391} a^{3} - \frac{868326141}{4361913391} a^{2} + \frac{2170320498}{4361913391} a - \frac{153212638}{396537581}$, $\frac{1}{5980066284262462985763386590} a^{15} + \frac{21333989273596822}{2990033142131231492881693295} a^{14} + \frac{341164577271223983527098}{2990033142131231492881693295} a^{13} - \frac{9547184898633788479042177}{5980066284262462985763386590} a^{12} + \frac{28429479954253657496129294}{2990033142131231492881693295} a^{11} - \frac{912414863872381602823532}{157370165375327973309562805} a^{10} - \frac{9482507804310862213603043}{2990033142131231492881693295} a^{9} - \frac{607609808029879727910646549}{2990033142131231492881693295} a^{8} + \frac{569813271583462507976898197}{5980066284262462985763386590} a^{7} - \frac{938381991321098070778669562}{2990033142131231492881693295} a^{6} + \frac{111104013339869343827840869}{271821194739202862989244845} a^{5} - \frac{656660405866254581709588079}{5980066284262462985763386590} a^{4} - \frac{260471211952163242130825031}{598006628426246298576338659} a^{3} - \frac{42969859084823350872814988}{598006628426246298576338659} a^{2} + \frac{16596448639293620900135274}{598006628426246298576338659} a + \frac{11569164397377228281875773}{54364238947840572597848969}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{12}\times C_{816}$, which has order $117504$ (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:  \( 54685.79274709562 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-85}) \), \(\Q(\sqrt{-255}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{3}, \sqrt{-85})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-17})\), \(\Q(\sqrt{5}, \sqrt{-17})\), \(\Q(\sqrt{15}, \sqrt{-51})\), \(\Q(\sqrt{5}, \sqrt{-51})\), \(\Q(\sqrt{15}, \sqrt{-17})\), 4.4.8000.1, 4.4.72000.1, 4.0.2312000.1, 4.0.20808000.2, 8.0.1082432160000.5, 8.8.82944000000.1, 8.0.6927565824000000.98, 8.0.85525504000000.30, 8.0.6927565824000000.89, 8.0.432972864000000.105, 8.0.432972864000000.123

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.16.2$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 20$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.2$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 20$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$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}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$