↑

Properties

Label 16.0.17153639972...2281.1
Degree $16$
Signature $[0, 8]$
Discriminant $7^{8}\cdot 29^{14}$
Root discriminant $50.37$
Ramified primes $7, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4025, -3780, 19427, -15783, 18236, -5728, -665, 2467, -1641, 700, -79, 25, 36, -21, 11, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 11*x^14 - 21*x^13 + 36*x^12 + 25*x^11 - 79*x^10 + 700*x^9 - 1641*x^8 + 2467*x^7 - 665*x^6 - 5728*x^5 + 18236*x^4 - 15783*x^3 + 19427*x^2 - 3780*x + 4025)
 
gp: K = bnfinit(x^16 - 4*x^15 + 11*x^14 - 21*x^13 + 36*x^12 + 25*x^11 - 79*x^10 + 700*x^9 - 1641*x^8 + 2467*x^7 - 665*x^6 - 5728*x^5 + 18236*x^4 - 15783*x^3 + 19427*x^2 - 3780*x + 4025, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 11 x^{14} - 21 x^{13} + 36 x^{12} + 25 x^{11} - 79 x^{10} + 700 x^{9} - 1641 x^{8} + 2467 x^{7} - 665 x^{6} - 5728 x^{5} + 18236 x^{4} - 15783 x^{3} + 19427 x^{2} - 3780 x + 4025 \)

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:  \(1715363997287681422874732281=7^{8}\cdot 29^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.37$
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:  $7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{10} a^{12} + \frac{3}{10} a^{10} + \frac{2}{5} a^{9} + \frac{3}{10} a^{8} + \frac{2}{5} a^{7} - \frac{3}{10} a^{5} + \frac{2}{5} a^{4} - \frac{3}{10} a^{3} + \frac{1}{5} a^{2} + \frac{1}{10} a - \frac{1}{2}$, $\frac{1}{530} a^{13} + \frac{2}{265} a^{12} - \frac{23}{530} a^{11} + \frac{121}{265} a^{10} - \frac{37}{106} a^{9} + \frac{123}{265} a^{8} + \frac{39}{265} a^{7} + \frac{151}{530} a^{6} - \frac{111}{265} a^{5} - \frac{49}{106} a^{4} + \frac{122}{265} a^{3} + \frac{143}{530} a^{2} - \frac{47}{530} a + \frac{7}{53}$, $\frac{1}{126670} a^{14} - \frac{38}{63335} a^{13} - \frac{2954}{63335} a^{12} - \frac{1079}{63335} a^{11} + \frac{6022}{12667} a^{10} + \frac{28458}{63335} a^{9} + \frac{23593}{126670} a^{8} + \frac{60691}{126670} a^{7} - \frac{17016}{63335} a^{6} - \frac{1349}{12667} a^{5} + \frac{13897}{63335} a^{4} - \frac{27841}{63335} a^{3} + \frac{36213}{126670} a^{2} - \frac{2997}{25334} a - \frac{9865}{25334}$, $\frac{1}{18663296904267557081410} a^{15} - \frac{8471801233345753}{9331648452133778540705} a^{14} - \frac{5086389739203279713}{9331648452133778540705} a^{13} + \frac{32683368096426739674}{1866329690426755708141} a^{12} + \frac{208855046668592994736}{9331648452133778540705} a^{11} + \frac{2868870017914675795776}{9331648452133778540705} a^{10} - \frac{17431591743489473413}{3732659380853511416282} a^{9} + \frac{52611086120510725101}{224858998846597073270} a^{8} - \frac{2740046698263802783541}{9331648452133778540705} a^{7} - \frac{8075263079796469940}{22485899884659707327} a^{6} - \frac{4448204975693013373539}{9331648452133778540705} a^{5} - \frac{3015534446426321952699}{9331648452133778540705} a^{4} - \frac{1820331435546026253751}{18663296904267557081410} a^{3} + \frac{8399924809873051762959}{18663296904267557081410} a^{2} + \frac{6812552564501121719499}{18663296904267557081410} a - \frac{923533047630219632065}{1866329690426755708141}$

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:  $7$
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:  \( 9780917.77015 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{-203}) \), 4.4.1195061.1, 4.0.24389.1, \(\Q(\sqrt{-7}, \sqrt{29})\), 8.4.41416953017909.1 x2, 8.0.845243939141.1 x2, 8.0.1428170793721.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
7.2.1.2$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}$
$29$29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$