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Properties

Label 16.0.86234126507...6096.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{12}\cdot 7^{8}$
Root discriminant $48.25$
Ramified primes $2, 3, 7$
Class number $200$ (GRH)
Class group $[2, 10, 10]$ (GRH)
Galois group $D_8$ (as 16T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![51937, 44064, 76792, -12288, 60292, -22224, 12208, -9984, 973, 624, 340, -168, 70, -24, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 - 24*x^13 + 70*x^12 - 168*x^11 + 340*x^10 + 624*x^9 + 973*x^8 - 9984*x^7 + 12208*x^6 - 22224*x^5 + 60292*x^4 - 12288*x^3 + 76792*x^2 + 44064*x + 51937)
 
gp: K = bnfinit(x^16 + 4*x^14 - 24*x^13 + 70*x^12 - 168*x^11 + 340*x^10 + 624*x^9 + 973*x^8 - 9984*x^7 + 12208*x^6 - 22224*x^5 + 60292*x^4 - 12288*x^3 + 76792*x^2 + 44064*x + 51937, 1)
 

Normalized defining polynomial

\( x^{16} + 4 x^{14} - 24 x^{13} + 70 x^{12} - 168 x^{11} + 340 x^{10} + 624 x^{9} + 973 x^{8} - 9984 x^{7} + 12208 x^{6} - 22224 x^{5} + 60292 x^{4} - 12288 x^{3} + 76792 x^{2} + 44064 x + 51937 \)

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:  \(862341265079199274522116096=2^{48}\cdot 3^{12}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.25$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{5869488243} a^{14} - \frac{978026224}{5869488243} a^{13} - \frac{36856114}{5869488243} a^{12} + \frac{212568862}{5869488243} a^{11} - \frac{237259078}{1956496081} a^{10} + \frac{506800747}{5869488243} a^{9} - \frac{97568128}{1956496081} a^{8} - \frac{31539155}{85065047} a^{7} + \frac{212741489}{1956496081} a^{6} + \frac{2441885639}{5869488243} a^{5} - \frac{209884452}{1956496081} a^{4} - \frac{66166646}{255195141} a^{3} + \frac{1036004117}{5869488243} a^{2} + \frac{2668018651}{5869488243} a - \frac{2907209726}{5869488243}$, $\frac{1}{25971292207343623783065559779} a^{15} + \frac{1088468144275413673}{25971292207343623783065559779} a^{14} + \frac{2264951454592556033603439740}{25971292207343623783065559779} a^{13} + \frac{340960720872942996805256459}{8657097402447874594355186593} a^{12} + \frac{1297609655606801525882902796}{8657097402447874594355186593} a^{11} - \frac{478445271856801694758393983}{8657097402447874594355186593} a^{10} + \frac{397797172654133593869765859}{8657097402447874594355186593} a^{9} + \frac{3049673136029432705573475164}{25971292207343623783065559779} a^{8} + \frac{510085898802174858875303031}{8657097402447874594355186593} a^{7} + \frac{4670063910614616410359940243}{25971292207343623783065559779} a^{6} + \frac{3849442852272283752050428111}{8657097402447874594355186593} a^{5} - \frac{1061009468432797999041237480}{8657097402447874594355186593} a^{4} + \frac{12490880411304723911694244382}{25971292207343623783065559779} a^{3} + \frac{3545109990131001134461663919}{8657097402447874594355186593} a^{2} - \frac{2831869791166173123547740458}{25971292207343623783065559779} a - \frac{10498698109255796023314823490}{25971292207343623783065559779}$

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

Class group and class number

$C_{2}\times C_{10}\times C_{10}$, which has order $200$ (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:  \( 32719.5368105 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times Q_8$ (as 16T7):

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 $D_8$
Character table for $D_8$

Intermediate fields

\(\Q(\sqrt{42}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{6}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{14})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{6}, \sqrt{14})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.12745506816.1, 8.0.12230590464.1, 8.0.29365647704064.1

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

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$