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Properties

Label 16.0.68457752166...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 5^{14}\cdot 101^{2}$
Root discriminant $41.18$
Ramified primes $2, 5, 101$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T1162

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![224401, -326000, 45836, 90400, 17132, -43980, -4332, 10640, 2025, -2020, -668, 320, 122, -20, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 16*x^14 - 20*x^13 + 122*x^12 + 320*x^11 - 668*x^10 - 2020*x^9 + 2025*x^8 + 10640*x^7 - 4332*x^6 - 43980*x^5 + 17132*x^4 + 90400*x^3 + 45836*x^2 - 326000*x + 224401)
 
gp: K = bnfinit(x^16 - 16*x^14 - 20*x^13 + 122*x^12 + 320*x^11 - 668*x^10 - 2020*x^9 + 2025*x^8 + 10640*x^7 - 4332*x^6 - 43980*x^5 + 17132*x^4 + 90400*x^3 + 45836*x^2 - 326000*x + 224401, 1)
 

Normalized defining polynomial

\( x^{16} - 16 x^{14} - 20 x^{13} + 122 x^{12} + 320 x^{11} - 668 x^{10} - 2020 x^{9} + 2025 x^{8} + 10640 x^{7} - 4332 x^{6} - 43980 x^{5} + 17132 x^{4} + 90400 x^{3} + 45836 x^{2} - 326000 x + 224401 \)

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:  \(68457752166400000000000000=2^{40}\cdot 5^{14}\cdot 101^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.18$
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:  $2, 5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{19} a^{14} + \frac{7}{19} a^{13} - \frac{3}{19} a^{12} - \frac{8}{19} a^{11} + \frac{3}{19} a^{10} + \frac{2}{19} a^{9} - \frac{2}{19} a^{8} + \frac{3}{19} a^{7} + \frac{9}{19} a^{6} - \frac{7}{19} a^{5} + \frac{7}{19} a^{4} + \frac{2}{19} a^{3} + \frac{3}{19} a^{2} + \frac{4}{19} a + \frac{4}{19}$, $\frac{1}{10648778387026635474720728721841} a^{15} + \frac{63714681140011599914153809179}{10648778387026635474720728721841} a^{14} - \frac{1331843542990210770707964928863}{10648778387026635474720728721841} a^{13} - \frac{43328028726890592120367491584}{10648778387026635474720728721841} a^{12} + \frac{1210635642412009206347601436224}{10648778387026635474720728721841} a^{11} + \frac{2844705583895202640650091981710}{10648778387026635474720728721841} a^{10} + \frac{4491539460040450958540375413784}{10648778387026635474720728721841} a^{9} - \frac{2353368176572200047958863595408}{10648778387026635474720728721841} a^{8} - \frac{3022256758429384361572756117455}{10648778387026635474720728721841} a^{7} + \frac{4985604999248595642853349367174}{10648778387026635474720728721841} a^{6} + \frac{36400253606702190507669044338}{10648778387026635474720728721841} a^{5} + \frac{3423152657548986895697520946037}{10648778387026635474720728721841} a^{4} - \frac{3173572281180307410929654534053}{10648778387026635474720728721841} a^{3} + \frac{173328783454741188114911073592}{10648778387026635474720728721841} a^{2} - \frac{2887969783737803399910879604562}{10648778387026635474720728721841} a + \frac{4198064949610369193753154039438}{10648778387026635474720728721841}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 501653.606929 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1162:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 49 conjugacy class representatives for t16n1162
Character table for t16n1162 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.0.5120000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
5Data not computed
$101$101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$