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Properties

Label 16.12.3104943754...0000.1
Degree $16$
Signature $[12, 2]$
Discriminant $2^{22}\cdot 5^{8}\cdot 17^{14}\cdot 103^{4}$
Root discriminant $220.42$
Ramified primes $2, 5, 17, 103$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T1392

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![75759616, 0, -152702976, 0, 24747648, 0, 9645664, 0, -1956632, 0, 59126, 0, 2448, 0, -119, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 119*x^14 + 2448*x^12 + 59126*x^10 - 1956632*x^8 + 9645664*x^6 + 24747648*x^4 - 152702976*x^2 + 75759616)
 
gp: K = bnfinit(x^16 - 119*x^14 + 2448*x^12 + 59126*x^10 - 1956632*x^8 + 9645664*x^6 + 24747648*x^4 - 152702976*x^2 + 75759616, 1)
 

Normalized defining polynomial

\( x^{16} - 119 x^{14} + 2448 x^{12} + 59126 x^{10} - 1956632 x^{8} + 9645664 x^{6} + 24747648 x^{4} - 152702976 x^{2} + 75759616 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(31049437544654067124090386841600000000=2^{22}\cdot 5^{8}\cdot 17^{14}\cdot 103^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $220.42$
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:  $2, 5, 17, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{136} a^{8} + \frac{1}{8} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{272} a^{9} + \frac{1}{16} a^{7} - \frac{1}{2} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{544} a^{10} + \frac{1}{544} a^{8} - \frac{1}{4} a^{6} + \frac{3}{16} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{2176} a^{11} - \frac{3}{2176} a^{9} - \frac{3}{32} a^{7} - \frac{13}{64} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{13056} a^{12} - \frac{11}{13056} a^{10} + \frac{11}{3264} a^{8} - \frac{77}{384} a^{6} + \frac{7}{48} a^{4} + \frac{3}{8} a^{2} - \frac{1}{3}$, $\frac{1}{52224} a^{13} - \frac{1}{26112} a^{12} - \frac{11}{52224} a^{11} - \frac{13}{26112} a^{10} - \frac{13}{13056} a^{9} - \frac{1}{384} a^{8} - \frac{125}{1536} a^{7} + \frac{173}{768} a^{6} - \frac{41}{192} a^{5} - \frac{1}{6} a^{4} + \frac{13}{32} a^{3} - \frac{1}{16} a^{2} - \frac{1}{12} a - \frac{1}{3}$, $\frac{1}{191326678801487794176} a^{14} - \frac{657068927307917}{63775559600495931392} a^{12} + \frac{2294522148142433}{5978958712546493568} a^{10} + \frac{311004720659315995}{95663339400743897088} a^{8} - \frac{195702459472073111}{1406813814716822016} a^{6} + \frac{88973579361864163}{351703453679205504} a^{4} + \frac{12049030681161601}{87925863419801376} a^{2} - \frac{1000205395974869}{2747683231868793}$, $\frac{1}{1530613430411902353408} a^{15} + \frac{12683103788043145}{1530613430411902353408} a^{13} + \frac{16495671634382969}{95663339400743897088} a^{11} + \frac{105844372679779451}{765306715205951176704} a^{9} + \frac{1104867603612488909}{11254510517734576128} a^{7} - \frac{107115705532402975}{937875876477881344} a^{5} - \frac{1}{2} a^{4} - \frac{42904633956214259}{703406907358411008} a^{3} + \frac{2108983296336193}{7327155284983448} a$

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:  $13$
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:  \( 258315808445000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1392:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 80 conjugacy class representatives for t16n1392 are not computed
Character table for t16n1392 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.122825.1, 4.4.2024156.1, 4.4.2976700.1, 8.8.2560754695210000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.8.6$x^{4} + 6 x^{2} + 4 x + 2$$4$$1$$8$$D_{4}$$[2, 3]^{2}$
2.4.9.3$x^{4} + 6 x^{2} + 10$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17Data not computed
$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}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$