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Properties

Label 16.0.51078407858...7216.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 151^{8}\cdot 229^{4}$
Root discriminant $227.39$
Ramified primes $2, 151, 229$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group 16T1049

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2607940624, 0, 1556049088, 0, 431686272, 0, 66741728, 0, 6001288, 0, 315376, 0, 9504, 0, 152, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 152*x^14 + 9504*x^12 + 315376*x^10 + 6001288*x^8 + 66741728*x^6 + 431686272*x^4 + 1556049088*x^2 + 2607940624)
 
gp: K = bnfinit(x^16 + 152*x^14 + 9504*x^12 + 315376*x^10 + 6001288*x^8 + 66741728*x^6 + 431686272*x^4 + 1556049088*x^2 + 2607940624, 1)
 

Normalized defining polynomial

\( x^{16} + 152 x^{14} + 9504 x^{12} + 315376 x^{10} + 6001288 x^{8} + 66741728 x^{6} + 431686272 x^{4} + 1556049088 x^{2} + 2607940624 \)

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:  \(51078407858373713535118085896290697216=2^{36}\cdot 151^{8}\cdot 229^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $227.39$
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:  $2, 151, 229$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{8} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{8} - \frac{1}{16} a^{6} + \frac{1}{8} a^{2} - \frac{3}{16}$, $\frac{1}{64} a^{9} - \frac{1}{16} a^{7} + \frac{1}{8} a^{3} - \frac{3}{16} a$, $\frac{1}{64} a^{10} + \frac{1}{16} a^{2}$, $\frac{1}{64} a^{11} + \frac{1}{16} a^{3}$, $\frac{1}{512} a^{12} + \frac{1}{256} a^{10} + \frac{1}{256} a^{8} - \frac{1}{16} a^{6} - \frac{7}{128} a^{4} + \frac{17}{64} a^{2} - \frac{23}{64}$, $\frac{1}{512} a^{13} + \frac{1}{256} a^{11} + \frac{1}{256} a^{9} - \frac{1}{16} a^{7} - \frac{7}{128} a^{5} + \frac{17}{64} a^{3} - \frac{23}{64} a$, $\frac{1}{6105890100371968} a^{14} - \frac{117186053223}{763236262546496} a^{12} + \frac{14939883448491}{3052945050185984} a^{10} - \frac{8769977655163}{1526472525092992} a^{8} + \frac{32745750170673}{1526472525092992} a^{6} + \frac{3681646014153}{190809065636624} a^{4} - \frac{17613842452485}{763236262546496} a^{2} + \frac{125669193819093}{381618131273248}$, $\frac{1}{77953898911448915456} a^{15} - \frac{2970403572395745}{77953898911448915456} a^{13} - \frac{82523413729416779}{19488474727862228864} a^{11} - \frac{134502154529323379}{38976949455724457728} a^{9} - \frac{981298278818986559}{19488474727862228864} a^{7} + \frac{2055034506647831}{1146380866344836992} a^{5} - \frac{1159036736647790719}{4872118681965557216} a^{3} - \frac{4495269875844234699}{9744237363931114432} a$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 901511177354 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1049:

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 t16n1049
Character table for t16n1049 is not computed

Intermediate fields

\(\Q(\sqrt{151}) \), 4.0.229.1, 8.0.6979410125322496.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
151Data not computed
229Data not computed