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Properties

Label 16.2.29288304000...0000.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,2^{16}\cdot 3^{10}\cdot 5^{12}\cdot 31$
Root discriminant $16.47$
Ramified primes $2, 3, 5, 31$
Class number $1$
Class group Trivial
Galois group 16T1616

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 2, -58, 241, -366, 215, 20, -104, 40, 20, -18, 4, -2, 5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 5*x^14 - 2*x^13 + 4*x^12 - 18*x^11 + 20*x^10 + 40*x^9 - 104*x^8 + 20*x^7 + 215*x^6 - 366*x^5 + 241*x^4 - 58*x^3 + 2*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 5*x^14 - 2*x^13 + 4*x^12 - 18*x^11 + 20*x^10 + 40*x^9 - 104*x^8 + 20*x^7 + 215*x^6 - 366*x^5 + 241*x^4 - 58*x^3 + 2*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 5 x^{14} - 2 x^{13} + 4 x^{12} - 18 x^{11} + 20 x^{10} + 40 x^{9} - 104 x^{8} + 20 x^{7} + 215 x^{6} - 366 x^{5} + 241 x^{4} - 58 x^{3} + 2 x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-29288304000000000000=-\,2^{16}\cdot 3^{10}\cdot 5^{12}\cdot 31\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.47$
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:  $2, 3, 5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{15} a^{14} - \frac{1}{15} a^{13} + \frac{1}{15} a^{12} + \frac{2}{15} a^{11} - \frac{2}{5} a^{10} + \frac{7}{15} a^{9} + \frac{2}{15} a^{8} - \frac{2}{5} a^{7} - \frac{4}{15} a^{6} - \frac{1}{15} a^{5} + \frac{2}{5} a^{4} - \frac{2}{15} a^{3} + \frac{4}{15} a^{2} + \frac{1}{15} a + \frac{1}{15}$, $\frac{1}{10764519735} a^{15} - \frac{195778442}{10764519735} a^{14} + \frac{136170412}{10764519735} a^{13} + \frac{107165647}{3588173245} a^{12} + \frac{1520011874}{3588173245} a^{11} + \frac{4289994058}{10764519735} a^{10} + \frac{1042749863}{2152903947} a^{9} + \frac{272571324}{3588173245} a^{8} + \frac{4350907702}{10764519735} a^{7} + \frac{1461328366}{3588173245} a^{6} + \frac{72875644}{3588173245} a^{5} + \frac{1003102042}{10764519735} a^{4} + \frac{220911562}{3588173245} a^{3} - \frac{4731877858}{10764519735} a^{2} + \frac{9326603}{717634649} a - \frac{639026107}{3588173245}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2444.13775743 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1616:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 94 conjugacy class representatives for t16n1616 are not computed
Character table for t16n1616 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 4.2.2000.1, 4.2.3600.1, 8.4.324000000.3

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
31Data not computed