↑

Properties

Label 16.0.76127245922...0625.4
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 89^{10}$
Root discriminant $55.28$
Ramified primes $5, 89$
Class number $740$ (GRH)
Class group $[740]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6561, -32805, 172044, -335340, 509814, -423225, 316503, -117105, 67004, -9895, 13121, 170, 1066, -15, 42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 42*x^14 - 15*x^13 + 1066*x^12 + 170*x^11 + 13121*x^10 - 9895*x^9 + 67004*x^8 - 117105*x^7 + 316503*x^6 - 423225*x^5 + 509814*x^4 - 335340*x^3 + 172044*x^2 - 32805*x + 6561)
 
gp: K = bnfinit(x^16 + 42*x^14 - 15*x^13 + 1066*x^12 + 170*x^11 + 13121*x^10 - 9895*x^9 + 67004*x^8 - 117105*x^7 + 316503*x^6 - 423225*x^5 + 509814*x^4 - 335340*x^3 + 172044*x^2 - 32805*x + 6561, 1)
 

Normalized defining polynomial

\( x^{16} + 42 x^{14} - 15 x^{13} + 1066 x^{12} + 170 x^{11} + 13121 x^{10} - 9895 x^{9} + 67004 x^{8} - 117105 x^{7} + 316503 x^{6} - 423225 x^{5} + 509814 x^{4} - 335340 x^{3} + 172044 x^{2} - 32805 x + 6561 \)

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:  \(7612724592276900293212890625=5^{12}\cdot 89^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.28$
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:  $5, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{18} a^{10} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{2}{9} a^{6} - \frac{1}{18} a^{5} - \frac{1}{18} a^{4} - \frac{2}{9} a^{3} - \frac{1}{18} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{54} a^{11} - \frac{1}{18} a^{9} + \frac{2}{9} a^{8} - \frac{7}{27} a^{7} - \frac{19}{54} a^{6} - \frac{19}{54} a^{5} - \frac{11}{27} a^{4} - \frac{19}{54} a^{3} - \frac{5}{18} a^{2} - \frac{1}{2} a$, $\frac{1}{162} a^{12} - \frac{1}{54} a^{10} + \frac{2}{27} a^{9} - \frac{7}{81} a^{8} + \frac{35}{162} a^{7} - \frac{19}{162} a^{6} - \frac{11}{81} a^{5} + \frac{35}{162} a^{4} - \frac{23}{54} a^{3} - \frac{1}{6} a^{2}$, $\frac{1}{486} a^{13} - \frac{1}{162} a^{11} + \frac{2}{81} a^{10} - \frac{7}{243} a^{9} + \frac{35}{486} a^{8} - \frac{19}{486} a^{7} - \frac{92}{243} a^{6} + \frac{35}{486} a^{5} - \frac{77}{162} a^{4} + \frac{5}{18} a^{3} - \frac{1}{3} a$, $\frac{1}{1458} a^{14} - \frac{1}{486} a^{12} + \frac{2}{243} a^{11} - \frac{7}{729} a^{10} + \frac{35}{1458} a^{9} - \frac{505}{1458} a^{8} - \frac{92}{729} a^{7} + \frac{521}{1458} a^{6} + \frac{85}{486} a^{5} + \frac{5}{54} a^{4} + \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{124209768938047084361278564862814} a^{15} - \frac{306914728398093640341152212}{6900542718780393575626586936823} a^{14} - \frac{6786431277908978772470963630}{20701628156341180726879760810469} a^{13} + \frac{252458443836658629393219020}{20701628156341180726879760810469} a^{12} + \frac{187155285809761594841863445642}{62104884469023542180639282431407} a^{11} - \frac{1054292871930328674589295639845}{124209768938047084361278564862814} a^{10} - \frac{1183256043727190378449662162950}{62104884469023542180639282431407} a^{9} - \frac{57141175958159238058591901944687}{124209768938047084361278564862814} a^{8} - \frac{29269862886213938245015892389928}{62104884469023542180639282431407} a^{7} + \frac{6494241042195297499819958004947}{20701628156341180726879760810469} a^{6} + \frac{5587763092345098887861937860317}{13801085437560787151253173873646} a^{5} - \frac{597671603332155168089614744435}{1533453937506754127917019319294} a^{4} + \frac{29932584769939110135580640891}{85191885417041895995389962183} a^{3} - \frac{59692990020246738038578959595}{170383770834083791990779924366} a^{2} + \frac{28294570781379188180759792189}{170383770834083791990779924366} a - \frac{13279490820281927175334042788}{28397295139013965331796654061}$

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

Class group and class number

$C_{740}$, which has order $740$ (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:  \( 24104.2464739 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, 4.0.990125.2, 4.0.11125.1, 8.0.3490037155625.1, 8.8.11015140625.1, 8.0.980347515625.3

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
5Data not computed
$89$89.4.3.4$x^{4} + 2403$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.4$x^{4} + 2403$$4$$1$$3$$C_4$$[\ ]_{4}$
89.8.4.1$x^{8} + 427734 x^{4} - 704969 x^{2} + 45739093689$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$