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Properties

Label 16.0.13647448497...6881.2
Degree $16$
Signature $[0, 8]$
Discriminant $17^{8}\cdot 89^{14}$
Root discriminant $209.38$
Ramified primes $17, 89$
Class number $10485034$ (GRH)
Class group $[10485034]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1870659584, 1321467904, 492929024, -177130496, 45164672, 54143648, -29626320, -10128816, 7486944, 1489066, -526969, -67142, 19155, 686, -235, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 235*x^14 + 686*x^13 + 19155*x^12 - 67142*x^11 - 526969*x^10 + 1489066*x^9 + 7486944*x^8 - 10128816*x^7 - 29626320*x^6 + 54143648*x^5 + 45164672*x^4 - 177130496*x^3 + 492929024*x^2 + 1321467904*x + 1870659584)
 
gp: K = bnfinit(x^16 - 2*x^15 - 235*x^14 + 686*x^13 + 19155*x^12 - 67142*x^11 - 526969*x^10 + 1489066*x^9 + 7486944*x^8 - 10128816*x^7 - 29626320*x^6 + 54143648*x^5 + 45164672*x^4 - 177130496*x^3 + 492929024*x^2 + 1321467904*x + 1870659584, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 235 x^{14} + 686 x^{13} + 19155 x^{12} - 67142 x^{11} - 526969 x^{10} + 1489066 x^{9} + 7486944 x^{8} - 10128816 x^{7} - 29626320 x^{6} + 54143648 x^{5} + 45164672 x^{4} - 177130496 x^{3} + 492929024 x^{2} + 1321467904 x + 1870659584 \)

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:  \(13647448497711210780660288977472356881=17^{8}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $209.38$
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:  $17, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{128} a^{8} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{128} a^{4} - \frac{1}{32} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{256} a^{9} + \frac{1}{128} a^{7} - \frac{15}{256} a^{5} + \frac{3}{64} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{256} a^{10} - \frac{1}{32} a^{7} - \frac{3}{256} a^{6} - \frac{3}{128} a^{4} - \frac{3}{32} a^{3} - \frac{7}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{1024} a^{11} - \frac{1}{1024} a^{10} - \frac{1}{512} a^{8} - \frac{19}{1024} a^{7} + \frac{15}{1024} a^{6} + \frac{21}{512} a^{5} + \frac{13}{256} a^{4} - \frac{3}{128} a^{3} - \frac{3}{16} a^{2} - \frac{1}{8} a$, $\frac{1}{2048} a^{12} - \frac{1}{2048} a^{11} - \frac{1}{512} a^{10} - \frac{1}{1024} a^{9} + \frac{5}{2048} a^{8} - \frac{49}{2048} a^{7} - \frac{13}{1024} a^{6} - \frac{3}{512} a^{5} - \frac{13}{256} a^{4} - \frac{5}{32} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8192} a^{13} + \frac{1}{8192} a^{11} - \frac{1}{2048} a^{10} + \frac{3}{8192} a^{9} - \frac{3}{1024} a^{8} - \frac{253}{8192} a^{7} + \frac{23}{2048} a^{6} - \frac{15}{1024} a^{5} + \frac{3}{64} a^{4} + \frac{119}{512} a^{3} + \frac{25}{128} a^{2} - \frac{7}{16} a$, $\frac{1}{65536} a^{14} - \frac{1}{16384} a^{13} + \frac{13}{65536} a^{12} - \frac{3}{16384} a^{11} - \frac{69}{65536} a^{10} - \frac{15}{16384} a^{9} + \frac{15}{65536} a^{8} + \frac{27}{16384} a^{7} + \frac{151}{8192} a^{6} - \frac{111}{2048} a^{5} + \frac{23}{4096} a^{4} + \frac{63}{1024} a^{3} - \frac{1}{64} a^{2} - \frac{7}{16} a$, $\frac{1}{396255298194972886770591427359863279813751996416} a^{15} - \frac{108873432710759649545654296574956857679225}{198127649097486443385295713679931639906875998208} a^{14} - \frac{1935976678377437101804461944156728224139585}{36023208926815716979144675214533025437613817856} a^{13} - \frac{40669994524227796760031854617467752739687169}{198127649097486443385295713679931639906875998208} a^{12} - \frac{119632766009362167605781924753526774535300621}{396255298194972886770591427359863279813751996416} a^{11} + \frac{204628863018252221651987987276482993213194613}{198127649097486443385295713679931639906875998208} a^{10} + \frac{174197261651328389938340421918356932224423143}{396255298194972886770591427359863279813751996416} a^{9} - \frac{89003956479664231154546113260504848703937139}{198127649097486443385295713679931639906875998208} a^{8} + \frac{9525453643531946805343995742439403953356493}{6191489034296451355790491052497863747089874944} a^{7} - \frac{524603612764030233169337792502432349849961611}{24765956137185805423161964209991454988359499776} a^{6} - \frac{1299059074100198556780090613481173663401479757}{24765956137185805423161964209991454988359499776} a^{5} + \frac{673499704276729937889880340468172051935905325}{12382978068592902711580982104995727494179749888} a^{4} - \frac{478395664937538502413691964784103818605079235}{3095744517148225677895245526248931873544937472} a^{3} - \frac{95428343042909734433837448164482904891651749}{386968064643528209736905690781116484193117184} a^{2} + \frac{3376597145190218579465880607272642818468311}{24185504040220513108556605673819780262069824} a + \frac{90217676940189354734204693076131352362583}{755797001256891034642393927306868133189682}$

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

Class group and class number

$C_{10485034}$, which has order $10485034$ (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:  \( 5387761504.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{89}) \), 4.0.11984473.1, 4.4.704969.1, 4.0.134657.1, 8.0.3694245321809477609.1, 8.8.12782855784807881.1, 8.0.143627593087729.1

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
89Data not computed