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Properties

Label 16.8.42868060426...8689.2
Degree $16$
Signature $[8, 4]$
Discriminant $37^{8}\cdot 73^{14}$
Root discriminant $259.72$
Ramified primes $37, 73$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1585081, 0, -9404606, 0, 14819405, 0, -12030478, 0, 4768268, 0, -486190, 0, -8235, 0, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 18*x^14 - 8235*x^12 - 486190*x^10 + 4768268*x^8 - 12030478*x^6 + 14819405*x^4 - 9404606*x^2 + 1585081)
 
gp: K = bnfinit(x^16 + 18*x^14 - 8235*x^12 - 486190*x^10 + 4768268*x^8 - 12030478*x^6 + 14819405*x^4 - 9404606*x^2 + 1585081, 1)
 

Normalized defining polynomial

\( x^{16} + 18 x^{14} - 8235 x^{12} - 486190 x^{10} + 4768268 x^{8} - 12030478 x^{6} + 14819405 x^{4} - 9404606 x^{2} + 1585081 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(428680604268109875973804253861404748689=37^{8}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $259.72$
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:  $37, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{6} - \frac{1}{8} a^{3} + \frac{1}{8}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{7} - \frac{1}{8} a^{4} + \frac{1}{8} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{8} a^{5} + \frac{1}{8} a^{2}$, $\frac{1}{32} a^{12} + \frac{1}{16} a^{6} - \frac{3}{32}$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{12} - \frac{1}{8} a^{8} + \frac{1}{32} a^{7} + \frac{3}{32} a^{6} - \frac{3}{8} a^{2} + \frac{29}{64} a - \frac{5}{64}$, $\frac{1}{28203497791500920253075584} a^{14} - \frac{390316358795463895234497}{28203497791500920253075584} a^{12} + \frac{175043394722098138309709}{3525437223937615031634448} a^{10} + \frac{216816736695150807538973}{14101748895750460126537792} a^{8} + \frac{562373746775633325019903}{14101748895750460126537792} a^{6} - \frac{386970588684701446378753}{3525437223937615031634448} a^{4} - \frac{12438856314648915155449083}{28203497791500920253075584} a^{2} - \frac{1}{2} a + \frac{12471241486539159784742243}{28203497791500920253075584}$, $\frac{1}{71016407438999317197244320512} a^{15} - \frac{1}{56406995583001840506151168} a^{14} - \frac{3034394276748675168960333}{71016407438999317197244320512} a^{13} - \frac{491042947188939862674115}{56406995583001840506151168} a^{12} + \frac{386210419415890944102281765}{8877050929874914649655540064} a^{11} - \frac{175043394722098138309709}{7050874447875230063268896} a^{10} - \frac{1840061414158739895705642883}{35508203719499658598622160256} a^{9} - \frac{216816736695150807538973}{28203497791500920253075584} a^{8} + \frac{3872373804936261341817552419}{35508203719499658598622160256} a^{7} + \frac{2081704171177577948705933}{28203497791500920253075584} a^{6} - \frac{1981682690441624349224938529}{8877050929874914649655540064} a^{5} - \frac{1375748023284106069438471}{7050874447875230063268896} a^{4} + \frac{9647259137274416271522938437}{71016407438999317197244320512} a^{3} + \frac{12438856314648915155449083}{56406995583001840506151168} a^{2} - \frac{2003197491299792234552253401}{71016407438999317197244320512} a - \frac{16878038016461178574285303}{56406995583001840506151168}$

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:  $11$
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:  \( 96118684976900 \) (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

\(\Q(\sqrt{73}) \), \(\Q(\sqrt{2701}) \), \(\Q(\sqrt{37}) \), 4.4.532564273.1, 4.4.389017.1, \(\Q(\sqrt{37}, \sqrt{73})\), 8.4.20704603455949352617.1 x2, 8.4.15123888572643793.1 x2, 8.8.283624704876018529.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
37Data not computed
$73$73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$