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Properties

Label 16.8.21943454004...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 19^{6}\cdot 103^{6}$
Root discriminant $38.36$
Ramified primes $5, 19, 103$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1048

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17936, -27208, -49184, -24504, -14617, -3964, 5236, 4407, 728, -60, 251, -70, -76, 11, -6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 6*x^14 + 11*x^13 - 76*x^12 - 70*x^11 + 251*x^10 - 60*x^9 + 728*x^8 + 4407*x^7 + 5236*x^6 - 3964*x^5 - 14617*x^4 - 24504*x^3 - 49184*x^2 - 27208*x + 17936)
 
gp: K = bnfinit(x^16 - 2*x^15 - 6*x^14 + 11*x^13 - 76*x^12 - 70*x^11 + 251*x^10 - 60*x^9 + 728*x^8 + 4407*x^7 + 5236*x^6 - 3964*x^5 - 14617*x^4 - 24504*x^3 - 49184*x^2 - 27208*x + 17936, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 6 x^{14} + 11 x^{13} - 76 x^{12} - 70 x^{11} + 251 x^{10} - 60 x^{9} + 728 x^{8} + 4407 x^{7} + 5236 x^{6} - 3964 x^{5} - 14617 x^{4} - 24504 x^{3} - 49184 x^{2} - 27208 x + 17936 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(21943454004015643378515625=5^{8}\cdot 19^{6}\cdot 103^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.36$
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:  $5, 19, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{103} a^{12} + \frac{18}{103} a^{11} - \frac{2}{103} a^{10} + \frac{37}{103} a^{9} - \frac{34}{103} a^{8} - \frac{10}{103} a^{7} - \frac{25}{103} a^{6} - \frac{45}{103} a^{5} - \frac{51}{103} a^{4} - \frac{27}{103} a^{3} + \frac{12}{103} a^{2} - \frac{36}{103} a + \frac{2}{103}$, $\frac{1}{206} a^{13} + \frac{43}{103} a^{11} + \frac{73}{206} a^{10} - \frac{41}{103} a^{9} - \frac{8}{103} a^{8} - \frac{51}{206} a^{7} + \frac{48}{103} a^{6} + \frac{19}{103} a^{5} + \frac{67}{206} a^{4} + \frac{43}{103} a^{3} - \frac{23}{103} a^{2} - \frac{71}{206} a - \frac{18}{103}$, $\frac{1}{7828} a^{14} - \frac{4}{1957} a^{13} - \frac{1}{3914} a^{12} - \frac{621}{7828} a^{11} - \frac{1361}{3914} a^{10} + \frac{1183}{3914} a^{9} + \frac{2579}{7828} a^{8} - \frac{31}{3914} a^{7} + \frac{845}{1957} a^{6} - \frac{3585}{7828} a^{5} + \frac{17}{38} a^{4} + \frac{805}{1957} a^{3} - \frac{597}{7828} a^{2} + \frac{31}{206} a - \frac{11}{103}$, $\frac{1}{55901653730001768855064} a^{15} - \frac{987809393500560605}{27950826865000884427532} a^{14} - \frac{23649571492019511417}{27950826865000884427532} a^{13} + \frac{244430135357901579499}{55901653730001768855064} a^{12} + \frac{6644813922337310006777}{13975413432500442213766} a^{11} + \frac{9291514893499117860871}{27950826865000884427532} a^{10} + \frac{10478711028080497157235}{55901653730001768855064} a^{9} - \frac{191540252835214144025}{735548075394760116514} a^{8} + \frac{195464787906847138995}{13975413432500442213766} a^{7} - \frac{17364969929438212898809}{55901653730001768855064} a^{6} + \frac{3510482078544713685}{735548075394760116514} a^{5} - \frac{2174381868945538748082}{6987706716250221106883} a^{4} + \frac{9050257922262820120991}{55901653730001768855064} a^{3} + \frac{800245656371980001297}{6987706716250221106883} a^{2} + \frac{332413271907732183981}{735548075394760116514} a + \frac{1556609896807655835}{6233458266057289123}$

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:  $11$
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:  \( 5617535.83163 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1048:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 768
The 31 conjugacy class representatives for t16n1048
Character table for t16n1048 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.1957.1, 8.8.2393655625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$103$103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.8.6.2$x^{8} + 927 x^{4} + 265225$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$