↑

Properties

Label 16.0.11819246862...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 3^{8}\cdot 5^{8}$
Root discriminant $56.82$
Ramified primes $2, 3, 5$
Class number $1764$ (GRH)
Class group $[42, 42]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1065439, -71320, 406796, -834912, 875066, -690440, 489568, -272568, 136411, -54408, 20368, -5944, 1658, -336, 68, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 68*x^14 - 336*x^13 + 1658*x^12 - 5944*x^11 + 20368*x^10 - 54408*x^9 + 136411*x^8 - 272568*x^7 + 489568*x^6 - 690440*x^5 + 875066*x^4 - 834912*x^3 + 406796*x^2 - 71320*x + 1065439)
 
gp: K = bnfinit(x^16 - 8*x^15 + 68*x^14 - 336*x^13 + 1658*x^12 - 5944*x^11 + 20368*x^10 - 54408*x^9 + 136411*x^8 - 272568*x^7 + 489568*x^6 - 690440*x^5 + 875066*x^4 - 834912*x^3 + 406796*x^2 - 71320*x + 1065439, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 68 x^{14} - 336 x^{13} + 1658 x^{12} - 5944 x^{11} + 20368 x^{10} - 54408 x^{9} + 136411 x^{8} - 272568 x^{7} + 489568 x^{6} - 690440 x^{5} + 875066 x^{4} - 834912 x^{3} + 406796 x^{2} - 71320 x + 1065439 \)

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:  \(11819246862071129702400000000=2^{62}\cdot 3^{8}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.82$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(480=2^{5}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{480}(1,·)$, $\chi_{480}(389,·)$, $\chi_{480}(269,·)$, $\chi_{480}(461,·)$, $\chi_{480}(149,·)$, $\chi_{480}(409,·)$, $\chi_{480}(221,·)$, $\chi_{480}(289,·)$, $\chi_{480}(101,·)$, $\chi_{480}(49,·)$, $\chi_{480}(361,·)$, $\chi_{480}(29,·)$, $\chi_{480}(241,·)$, $\chi_{480}(169,·)$, $\chi_{480}(121,·)$, $\chi_{480}(341,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{9} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{4} - \frac{1}{3} a^{2} - \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{2}{9}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{4}{9} a^{3} - \frac{4}{9} a^{2} + \frac{1}{9}$, $\frac{1}{27} a^{12} + \frac{1}{27} a^{9} - \frac{4}{27} a^{6} + \frac{8}{27} a^{3} + \frac{10}{27}$, $\frac{1}{27} a^{13} + \frac{1}{27} a^{10} - \frac{4}{27} a^{7} - \frac{1}{27} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{8}{27} a - \frac{1}{3}$, $\frac{1}{901382348507211} a^{14} - \frac{7}{901382348507211} a^{13} - \frac{7608699482581}{901382348507211} a^{12} + \frac{45652196895577}{901382348507211} a^{11} - \frac{40126288436926}{901382348507211} a^{10} - \frac{17539840801168}{901382348507211} a^{9} - \frac{4180496349871}{901382348507211} a^{8} - \frac{76939963891928}{901382348507211} a^{7} + \frac{23172152058952}{901382348507211} a^{6} - \frac{65591937508066}{901382348507211} a^{5} + \frac{37722940361791}{901382348507211} a^{4} - \frac{253798217235692}{901382348507211} a^{3} - \frac{192912421564115}{901382348507211} a^{2} - \frac{148924583996020}{901382348507211} a + \frac{437774465163476}{901382348507211}$, $\frac{1}{238268705857350634107} a^{15} + \frac{132161}{238268705857350634107} a^{14} + \frac{4187113707304144496}{238268705857350634107} a^{13} + \frac{888271532524355656}{79422901952450211369} a^{12} + \frac{8717635452483544052}{238268705857350634107} a^{11} + \frac{1333850016463272173}{238268705857350634107} a^{10} + \frac{1185630251799538027}{238268705857350634107} a^{9} + \frac{7700051387021267344}{238268705857350634107} a^{8} + \frac{26643677444706773446}{238268705857350634107} a^{7} - \frac{1718759418160442923}{26474300650816737123} a^{6} + \frac{25648311298740831895}{238268705857350634107} a^{5} - \frac{1084497104921880257}{238268705857350634107} a^{4} - \frac{84900521763289703482}{238268705857350634107} a^{3} + \frac{76361811516896138714}{238268705857350634107} a^{2} + \frac{56273458643376391400}{238268705857350634107} a + \frac{2808517079059906880}{7686087285720988197}$

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

Class group and class number

$C_{42}\times C_{42}$, which has order $1764$ (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:  \( 12198.951274811623 \) (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

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{16})^+\), 4.4.51200.1, 8.8.2621440000.1, 8.0.173946175488.1, 8.0.108716359680000.13

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
5Data not computed