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Properties

Label 16.4.81667851444...0625.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{6}\cdot 13^{6}\cdot 101^{8}$
Root discriminant $48.08$
Ramified primes $5, 13, 101$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T1276

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, 2050, -3679, -19771, -84, 25877, -11766, -4225, 4902, -2615, 1778, -1039, 387, -98, 25, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 25*x^14 - 98*x^13 + 387*x^12 - 1039*x^11 + 1778*x^10 - 2615*x^9 + 4902*x^8 - 4225*x^7 - 11766*x^6 + 25877*x^5 - 84*x^4 - 19771*x^3 - 3679*x^2 + 2050*x + 625)
 
gp: K = bnfinit(x^16 - 7*x^15 + 25*x^14 - 98*x^13 + 387*x^12 - 1039*x^11 + 1778*x^10 - 2615*x^9 + 4902*x^8 - 4225*x^7 - 11766*x^6 + 25877*x^5 - 84*x^4 - 19771*x^3 - 3679*x^2 + 2050*x + 625, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 25 x^{14} - 98 x^{13} + 387 x^{12} - 1039 x^{11} + 1778 x^{10} - 2615 x^{9} + 4902 x^{8} - 4225 x^{7} - 11766 x^{6} + 25877 x^{5} - 84 x^{4} - 19771 x^{3} - 3679 x^{2} + 2050 x + 625 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(816678514443119949906390625=5^{6}\cdot 13^{6}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.08$
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:  $5, 13, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} + \frac{5}{17} a^{13} - \frac{5}{17} a^{12} + \frac{4}{17} a^{11} + \frac{1}{17} a^{10} + \frac{7}{17} a^{9} + \frac{4}{17} a^{8} - \frac{1}{17} a^{7} + \frac{8}{17} a^{6} + \frac{7}{17} a^{5} + \frac{8}{17} a^{4} - \frac{4}{17} a^{3} - \frac{2}{17} a^{2} - \frac{4}{17} a + \frac{6}{17}$, $\frac{1}{1692170812349956631531654820975} a^{15} + \frac{21085805232764888870244418843}{1692170812349956631531654820975} a^{14} + \frac{5671835397822463919302663308}{22562277497999421753755397613} a^{13} - \frac{670834734289881289327715011298}{1692170812349956631531654820975} a^{12} - \frac{530924643056641937403048084163}{1692170812349956631531654820975} a^{11} - \frac{14212910677917145981993987738}{564056937449985543843884940325} a^{10} - \frac{14398956089410737063442429391}{99539459549997448913626754175} a^{9} + \frac{32141015728576389373959590602}{338434162469991326306330964195} a^{8} - \frac{139225613079539316291736721198}{1692170812349956631531654820975} a^{7} + \frac{895640440496630392859176637}{3981578381999897956545070167} a^{6} + \frac{397545478675394574259505114459}{1692170812349956631531654820975} a^{5} - \frac{76469533131470477359865523941}{564056937449985543843884940325} a^{4} + \frac{227462596198621288360132369697}{564056937449985543843884940325} a^{3} + \frac{453707273387881377624057241904}{1692170812349956631531654820975} a^{2} - \frac{122855671960617829792919905818}{564056937449985543843884940325} a - \frac{17520649742111902659543327059}{67686832493998265261266192839}$

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:  $9$
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:  \( 6231586.51202 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1276:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1276
Character table for t16n1276 is not computed

Intermediate fields

\(\Q(\sqrt{101}) \), 4.4.132613.1, 8.4.87931038845.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.3.4$x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
$101$101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$