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Properties

Label 16.0.29702751525...241.18
Degree $16$
Signature $[0, 8]$
Discriminant $23^{8}\cdot 41^{14}$
Root discriminant $123.61$
Ramified primes $23, 41$
Class number $384$ (GRH)
Class group $[2, 4, 48]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![51165007, 119692325, 32623454, -51785747, 36803076, -7639120, 592081, 250756, 111126, -42783, 10940, -190, 253, -6, 12, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 12*x^14 - 6*x^13 + 253*x^12 - 190*x^11 + 10940*x^10 - 42783*x^9 + 111126*x^8 + 250756*x^7 + 592081*x^6 - 7639120*x^5 + 36803076*x^4 - 51785747*x^3 + 32623454*x^2 + 119692325*x + 51165007)
 
gp: K = bnfinit(x^16 - 2*x^15 + 12*x^14 - 6*x^13 + 253*x^12 - 190*x^11 + 10940*x^10 - 42783*x^9 + 111126*x^8 + 250756*x^7 + 592081*x^6 - 7639120*x^5 + 36803076*x^4 - 51785747*x^3 + 32623454*x^2 + 119692325*x + 51165007, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 12 x^{14} - 6 x^{13} + 253 x^{12} - 190 x^{11} + 10940 x^{10} - 42783 x^{9} + 111126 x^{8} + 250756 x^{7} + 592081 x^{6} - 7639120 x^{5} + 36803076 x^{4} - 51785747 x^{3} + 32623454 x^{2} + 119692325 x + 51165007 \)

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:  \(2970275152580737243393920061992241=23^{8}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $123.61$
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:  $23, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{31} a^{8} + \frac{14}{31} a^{7} + \frac{1}{31} a^{6} + \frac{8}{31} a^{5} + \frac{5}{31} a^{4} + \frac{8}{31} a^{3} + \frac{12}{31} a^{2} - \frac{15}{31} a + \frac{12}{31}$, $\frac{1}{31} a^{9} - \frac{9}{31} a^{7} - \frac{6}{31} a^{6} - \frac{14}{31} a^{5} - \frac{7}{31} a^{3} + \frac{3}{31} a^{2} + \frac{5}{31} a - \frac{13}{31}$, $\frac{1}{31} a^{10} - \frac{4}{31} a^{7} - \frac{5}{31} a^{6} + \frac{10}{31} a^{5} + \frac{7}{31} a^{4} + \frac{13}{31} a^{3} - \frac{11}{31} a^{2} + \frac{7}{31} a + \frac{15}{31}$, $\frac{1}{31} a^{11} - \frac{11}{31} a^{7} + \frac{14}{31} a^{6} + \frac{8}{31} a^{5} + \frac{2}{31} a^{4} - \frac{10}{31} a^{3} - \frac{7}{31} a^{2} - \frac{14}{31} a - \frac{14}{31}$, $\frac{1}{1426} a^{12} + \frac{3}{713} a^{11} - \frac{10}{713} a^{10} + \frac{1}{1426} a^{9} + \frac{2}{713} a^{8} + \frac{68}{713} a^{7} + \frac{635}{1426} a^{6} - \frac{84}{713} a^{5} + \frac{139}{713} a^{4} - \frac{245}{1426} a^{3} + \frac{3}{713} a^{2} + \frac{236}{713} a + \frac{3}{46}$, $\frac{1}{1426} a^{13} - \frac{5}{713} a^{11} - \frac{17}{1426} a^{10} - \frac{1}{713} a^{9} + \frac{10}{713} a^{8} + \frac{3}{1426} a^{7} + \frac{58}{713} a^{6} - \frac{231}{713} a^{5} - \frac{395}{1426} a^{4} - \frac{44}{713} a^{3} + \frac{264}{713} a^{2} - \frac{117}{1426} a - \frac{49}{713}$, $\frac{1}{84134} a^{14} - \frac{3}{84134} a^{13} + \frac{9}{84134} a^{12} + \frac{357}{84134} a^{11} + \frac{727}{84134} a^{10} - \frac{1289}{84134} a^{9} + \frac{433}{84134} a^{8} + \frac{1651}{3658} a^{7} + \frac{13325}{84134} a^{6} - \frac{29111}{84134} a^{5} - \frac{33595}{84134} a^{4} + \frac{21667}{84134} a^{3} + \frac{575}{3658} a^{2} - \frac{535}{2714} a + \frac{8501}{84134}$, $\frac{1}{7941737229841455039167875020511280178638} a^{15} - \frac{108593132116556081936734057265572}{128092535965184758696256048717923873849} a^{14} + \frac{1133116951806274783779877022648669115}{7941737229841455039167875020511280178638} a^{13} + \frac{767844669446950769745459185438913101}{3970868614920727519583937510255640089319} a^{12} + \frac{50672352808336327190953438730493206414}{3970868614920727519583937510255640089319} a^{11} - \frac{36163647161289509583737122490096538389}{7941737229841455039167875020511280178638} a^{10} - \frac{45315745623612406796433474381280952571}{3970868614920727519583937510255640089319} a^{9} - \frac{21320877389967090082364112344243472803}{3970868614920727519583937510255640089319} a^{8} - \frac{3888339054319180654601378427423841443717}{7941737229841455039167875020511280178638} a^{7} + \frac{399864431228404734016685117392544909945}{3970868614920727519583937510255640089319} a^{6} + \frac{1081147247416584698763943990866090350548}{3970868614920727519583937510255640089319} a^{5} + \frac{2973094095987058762885631228759533113905}{7941737229841455039167875020511280178638} a^{4} + \frac{742655784062696105774959444107439782115}{3970868614920727519583937510255640089319} a^{3} + \frac{1211502986622624590140134701048306182022}{3970868614920727519583937510255640089319} a^{2} - \frac{2666891304393394226114097146607054171185}{7941737229841455039167875020511280178638} a - \frac{32922477595249036953937198945077439003}{345292923036585001702951087848316529506}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{48}$, which has order $384$ (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:  \( 46045060.8105 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-943}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-23}, \sqrt{41})\), 4.4.68921.1, 4.0.36459209.2, 8.0.1329273920905681.3, 8.4.103025010883049.3 x2

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$