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Properties

Label 16.4.44208220679...0000.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{36}\cdot 3^{6}\cdot 5^{8}\cdot 7^{4}\cdot 97^{2}$
Root discriminant $46.27$
Ramified primes $2, 3, 5, 7, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1547

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3649, -7496, 5164, -1936, 2936, -760, 12, 608, -514, 8, -348, -16, -40, -8, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 - 8*x^13 - 40*x^12 - 16*x^11 - 348*x^10 + 8*x^9 - 514*x^8 + 608*x^7 + 12*x^6 - 760*x^5 + 2936*x^4 - 1936*x^3 + 5164*x^2 - 7496*x + 3649)
 
gp: K = bnfinit(x^16 + 4*x^14 - 8*x^13 - 40*x^12 - 16*x^11 - 348*x^10 + 8*x^9 - 514*x^8 + 608*x^7 + 12*x^6 - 760*x^5 + 2936*x^4 - 1936*x^3 + 5164*x^2 - 7496*x + 3649, 1)
 

Normalized defining polynomial

\( x^{16} + 4 x^{14} - 8 x^{13} - 40 x^{12} - 16 x^{11} - 348 x^{10} + 8 x^{9} - 514 x^{8} + 608 x^{7} + 12 x^{6} - 760 x^{5} + 2936 x^{4} - 1936 x^{3} + 5164 x^{2} - 7496 x + 3649 \)

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:  \(442082206796061081600000000=2^{36}\cdot 3^{6}\cdot 5^{8}\cdot 7^{4}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.27$
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:  $2, 3, 5, 7, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{3}{8} a^{2} - \frac{3}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} + \frac{1}{8}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{16} a^{5} + \frac{1}{16} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{16} a - \frac{1}{16}$, $\frac{1}{1552} a^{14} - \frac{7}{776} a^{13} - \frac{43}{1552} a^{12} - \frac{39}{776} a^{11} + \frac{55}{1552} a^{10} - \frac{17}{388} a^{9} + \frac{43}{1552} a^{8} + \frac{27}{194} a^{7} - \frac{213}{1552} a^{6} - \frac{5}{776} a^{5} - \frac{81}{1552} a^{4} + \frac{335}{776} a^{3} - \frac{147}{1552} a^{2} + \frac{165}{388} a - \frac{559}{1552}$, $\frac{1}{7827314667853406608} a^{15} - \frac{587959616315957}{1956828666963351652} a^{14} - \frac{234304187825886431}{7827314667853406608} a^{13} - \frac{74888331010082207}{1956828666963351652} a^{12} + \frac{463316806281685679}{7827314667853406608} a^{11} + \frac{33714533111058107}{978414333481675826} a^{10} + \frac{443527124667510737}{7827314667853406608} a^{9} - \frac{325304548815168019}{3913657333926703304} a^{8} + \frac{1516634186448176467}{7827314667853406608} a^{7} - \frac{102816948817585013}{1956828666963351652} a^{6} - \frac{3728274939719367}{182030573671009456} a^{5} - \frac{76467103579577687}{1956828666963351652} a^{4} + \frac{1519012493598516149}{7827314667853406608} a^{3} - \frac{126113157304559481}{978414333481675826} a^{2} + \frac{2944551053152514571}{7827314667853406608} a + \frac{20053489366797063}{95455056925041544}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 7148314.22184 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1547:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 133 conjugacy class representatives for t16n1547 are not computed
Character table for t16n1547 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.18063360000.1

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 R R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$