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Properties

Label 16.4.56211678682...8576.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{16}\cdot 3^{8}\cdot 7^{8}\cdot 11^{2}\cdot 37^{4}$
Root discriminant $30.50$
Ramified primes $2, 3, 7, 11, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1701

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37, 148, 681, 1596, 416, -2070, -724, 1522, 95, -780, 214, 206, -130, -2, 23, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 23*x^14 - 2*x^13 - 130*x^12 + 206*x^11 + 214*x^10 - 780*x^9 + 95*x^8 + 1522*x^7 - 724*x^6 - 2070*x^5 + 416*x^4 + 1596*x^3 + 681*x^2 + 148*x + 37)
 
gp: K = bnfinit(x^16 - 8*x^15 + 23*x^14 - 2*x^13 - 130*x^12 + 206*x^11 + 214*x^10 - 780*x^9 + 95*x^8 + 1522*x^7 - 724*x^6 - 2070*x^5 + 416*x^4 + 1596*x^3 + 681*x^2 + 148*x + 37, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 23 x^{14} - 2 x^{13} - 130 x^{12} + 206 x^{11} + 214 x^{10} - 780 x^{9} + 95 x^{8} + 1522 x^{7} - 724 x^{6} - 2070 x^{5} + 416 x^{4} + 1596 x^{3} + 681 x^{2} + 148 x + 37 \)

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:  \(562116786825947096088576=2^{16}\cdot 3^{8}\cdot 7^{8}\cdot 11^{2}\cdot 37^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.50$
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:  $2, 3, 7, 11, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7} a^{11} - \frac{2}{7} a^{10} - \frac{3}{7} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{2} - \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{14} a^{12} - \frac{2}{7} a^{9} - \frac{5}{14} a^{8} + \frac{1}{7} a^{7} - \frac{3}{14} a^{6} - \frac{3}{14} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{2}{7} a + \frac{5}{14}$, $\frac{1}{14} a^{13} - \frac{2}{7} a^{10} - \frac{5}{14} a^{9} + \frac{1}{7} a^{8} - \frac{3}{14} a^{7} - \frac{3}{14} a^{5} + \frac{3}{7} a^{4} + \frac{2}{7} a^{3} + \frac{2}{7} a^{2} + \frac{5}{14} a$, $\frac{1}{28} a^{14} - \frac{1}{28} a^{13} - \frac{1}{28} a^{12} + \frac{5}{28} a^{10} + \frac{13}{28} a^{9} + \frac{2}{7} a^{8} + \frac{1}{4} a^{7} + \frac{5}{14} a^{6} + \frac{5}{28} a^{5} - \frac{5}{28} a^{4} - \frac{3}{14} a^{3} - \frac{1}{4} a^{2} + \frac{1}{28} a - \frac{9}{28}$, $\frac{1}{3802563853060} a^{15} - \frac{3775758953}{271611703790} a^{14} - \frac{3578122201}{950640963265} a^{13} - \frac{67149589}{77603343940} a^{12} + \frac{229867907069}{3802563853060} a^{11} - \frac{88435157831}{190128192653} a^{10} - \frac{1706752849281}{3802563853060} a^{9} + \frac{81791507887}{543223407580} a^{8} - \frac{764750364941}{3802563853060} a^{7} + \frac{261214274763}{543223407580} a^{6} + \frac{578333086291}{1901281926530} a^{5} + \frac{1622853836057}{3802563853060} a^{4} - \frac{650347975317}{3802563853060} a^{3} - \frac{108532417164}{950640963265} a^{2} + \frac{53771402383}{380256385306} a + \frac{686440904623}{3802563853060}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 1220922.74221 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1701:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 119 conjugacy class representatives for t16n1701 are not computed
Character table for t16n1701 is not computed

Intermediate fields

\(\Q(\sqrt{21}) \), 4.4.16317.1, 8.4.68158589184.11

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 ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ R R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.8.8.3$x^{8} + 2 x^{7} + 2 x^{6} + 16$$2$$4$$8$$C_2^3: C_4$$[2, 2, 2]^{4}$
2.8.8.3$x^{8} + 2 x^{7} + 2 x^{6} + 16$$2$$4$$8$$C_2^3: C_4$$[2, 2, 2]^{4}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$37$37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$