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Properties

Label 16.16.1213655108...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{44}\cdot 5^{12}\cdot 41^{4}$
Root discriminant $56.92$
Ramified primes $2, 5, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3$ (as 16T268)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-104959, -382980, -284700, 456648, 647172, -108060, -431036, -50748, 128879, 26292, -19156, -3804, 1544, 216, -64, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 64*x^14 + 216*x^13 + 1544*x^12 - 3804*x^11 - 19156*x^10 + 26292*x^9 + 128879*x^8 - 50748*x^7 - 431036*x^6 - 108060*x^5 + 647172*x^4 + 456648*x^3 - 284700*x^2 - 382980*x - 104959)
 
gp: K = bnfinit(x^16 - 4*x^15 - 64*x^14 + 216*x^13 + 1544*x^12 - 3804*x^11 - 19156*x^10 + 26292*x^9 + 128879*x^8 - 50748*x^7 - 431036*x^6 - 108060*x^5 + 647172*x^4 + 456648*x^3 - 284700*x^2 - 382980*x - 104959, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 64 x^{14} + 216 x^{13} + 1544 x^{12} - 3804 x^{11} - 19156 x^{10} + 26292 x^{9} + 128879 x^{8} - 50748 x^{7} - 431036 x^{6} - 108060 x^{5} + 647172 x^{4} + 456648 x^{3} - 284700 x^{2} - 382980 x - 104959 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12136551081312256000000000000=2^{44}\cdot 5^{12}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.92$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{15} a^{12} + \frac{1}{15} a^{11} - \frac{1}{15} a^{9} - \frac{1}{15} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{4}{15} a^{5} + \frac{1}{5} a^{4} + \frac{4}{15} a^{3} - \frac{1}{3} a^{2} - \frac{2}{5} a - \frac{7}{15}$, $\frac{1}{15} a^{13} - \frac{1}{15} a^{11} - \frac{1}{15} a^{10} - \frac{1}{15} a^{8} + \frac{4}{15} a^{7} + \frac{2}{15} a^{6} - \frac{1}{15} a^{5} - \frac{2}{15} a^{4} - \frac{2}{5} a^{3} - \frac{1}{15} a^{2} - \frac{1}{15} a + \frac{4}{15}$, $\frac{1}{75} a^{14} + \frac{2}{75} a^{12} - \frac{1}{75} a^{11} - \frac{1}{25} a^{10} - \frac{7}{75} a^{9} - \frac{2}{75} a^{8} - \frac{13}{75} a^{7} + \frac{7}{15} a^{6} - \frac{11}{75} a^{5} - \frac{1}{5} a^{4} - \frac{16}{75} a^{3} + \frac{8}{75} a^{2} + \frac{34}{75} a + \frac{3}{25}$, $\frac{1}{4551372187009128291540675} a^{15} - \frac{1447462471440435596386}{1517124062336376097180225} a^{14} - \frac{59803816470005196383678}{4551372187009128291540675} a^{13} - \frac{2687744733589478310139}{137920369303306917925475} a^{12} + \frac{1198805159539974553385}{182054887480365131661627} a^{11} - \frac{305073843156436642846}{1517124062336376097180225} a^{10} - \frac{141277365878182683308897}{1517124062336376097180225} a^{9} + \frac{118587506587939040554193}{4551372187009128291540675} a^{8} + \frac{880251028430405471536739}{4551372187009128291540675} a^{7} - \frac{1749094209926464416900206}{4551372187009128291540675} a^{6} + \frac{678272796020781759254063}{4551372187009128291540675} a^{5} - \frac{1409616442676734025261471}{4551372187009128291540675} a^{4} - \frac{420927521218809232790068}{1517124062336376097180225} a^{3} - \frac{118057890056952482181199}{303424812467275219436045} a^{2} + \frac{1951833468842350887746242}{4551372187009128291540675} a + \frac{2270961887775656299068628}{4551372187009128291540675}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 803814836.688 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T268):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{20})^+\), 4.4.8000.1, \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$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}$
41Data not computed