↑

Properties

Label 16.0.54419558400...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{14}$
Root discriminant $26.36$
Ramified primes $2, 3, 5$
Class number $8$
Class group $[2, 2, 2]$
Galois group $C_8: C_2$ (as 16T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1021, -4374, 9870, -15200, 16655, -12792, 7638, -3150, 2070, -1790, 1758, -1062, 480, -130, 30, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 30*x^14 - 130*x^13 + 480*x^12 - 1062*x^11 + 1758*x^10 - 1790*x^9 + 2070*x^8 - 3150*x^7 + 7638*x^6 - 12792*x^5 + 16655*x^4 - 15200*x^3 + 9870*x^2 - 4374*x + 1021)
 
gp: K = bnfinit(x^16 - 4*x^15 + 30*x^14 - 130*x^13 + 480*x^12 - 1062*x^11 + 1758*x^10 - 1790*x^9 + 2070*x^8 - 3150*x^7 + 7638*x^6 - 12792*x^5 + 16655*x^4 - 15200*x^3 + 9870*x^2 - 4374*x + 1021, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 30 x^{14} - 130 x^{13} + 480 x^{12} - 1062 x^{11} + 1758 x^{10} - 1790 x^{9} + 2070 x^{8} - 3150 x^{7} + 7638 x^{6} - 12792 x^{5} + 16655 x^{4} - 15200 x^{3} + 9870 x^{2} - 4374 x + 1021 \)

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:  \(54419558400000000000000=2^{24}\cdot 3^{12}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.36$
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 over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{5} + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{7} + \frac{1}{5} a^{2}$, $\frac{1}{145} a^{13} + \frac{4}{145} a^{12} + \frac{2}{145} a^{11} - \frac{1}{145} a^{10} - \frac{1}{29} a^{9} + \frac{12}{145} a^{8} + \frac{3}{145} a^{7} - \frac{46}{145} a^{6} - \frac{42}{145} a^{5} - \frac{3}{29} a^{4} + \frac{6}{145} a^{3} + \frac{39}{145} a^{2} + \frac{2}{145} a - \frac{36}{145}$, $\frac{1}{4205} a^{14} + \frac{11}{4205} a^{13} + \frac{349}{4205} a^{12} + \frac{71}{4205} a^{11} - \frac{43}{841} a^{10} - \frac{81}{4205} a^{9} + \frac{8}{145} a^{8} + \frac{1831}{4205} a^{7} + \frac{506}{4205} a^{6} + \frac{474}{4205} a^{5} + \frac{1003}{4205} a^{4} + \frac{1241}{4205} a^{3} + \frac{507}{4205} a^{2} - \frac{132}{841} a - \frac{1006}{4205}$, $\frac{1}{159492809111368609645} a^{15} - \frac{12198852223877191}{159492809111368609645} a^{14} - \frac{247556086101422821}{159492809111368609645} a^{13} - \frac{11946600911291456913}{159492809111368609645} a^{12} + \frac{3152908455059951776}{31898561822273721929} a^{11} - \frac{6208486550396567977}{159492809111368609645} a^{10} - \frac{2752179901240407472}{159492809111368609645} a^{9} - \frac{8738054506150959548}{159492809111368609645} a^{8} - \frac{28576878214772569194}{159492809111368609645} a^{7} - \frac{41713906770765971013}{159492809111368609645} a^{6} - \frac{295846719902895817}{5499752038323055505} a^{5} - \frac{12541746336670219741}{159492809111368609645} a^{4} + \frac{19666684093303142973}{159492809111368609645} a^{3} - \frac{8936062630923605571}{159492809111368609645} a^{2} - \frac{2639138445130304397}{5499752038323055505} a - \frac{5527653141610802436}{31898561822273721929}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3121.7160225 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{60})^+\), 8.0.14580000000.1 x2

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

Sibling fields

Degree 8 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 R R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
3Data not computed
5Data not computed