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Properties

Label 16.0.11638459724...7856.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 17^{4}\cdot 89^{2}$
Root discriminant $23.94$
Ramified primes $2, 17, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T799

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![578, 3400, 8728, 10928, 4602, -4340, -5164, -476, 1675, 584, -304, -152, 34, 28, -4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 4*x^14 + 28*x^13 + 34*x^12 - 152*x^11 - 304*x^10 + 584*x^9 + 1675*x^8 - 476*x^7 - 5164*x^6 - 4340*x^5 + 4602*x^4 + 10928*x^3 + 8728*x^2 + 3400*x + 578)
 
gp: K = bnfinit(x^16 - 4*x^15 - 4*x^14 + 28*x^13 + 34*x^12 - 152*x^11 - 304*x^10 + 584*x^9 + 1675*x^8 - 476*x^7 - 5164*x^6 - 4340*x^5 + 4602*x^4 + 10928*x^3 + 8728*x^2 + 3400*x + 578, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 4 x^{14} + 28 x^{13} + 34 x^{12} - 152 x^{11} - 304 x^{10} + 584 x^{9} + 1675 x^{8} - 476 x^{7} - 5164 x^{6} - 4340 x^{5} + 4602 x^{4} + 10928 x^{3} + 8728 x^{2} + 3400 x + 578 \)

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:  \(11638459724246712057856=2^{44}\cdot 17^{4}\cdot 89^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.94$
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:  $2, 17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} - \frac{2}{7} a^{12} - \frac{3}{7} a^{10} + \frac{2}{7} a^{9} + \frac{3}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} - \frac{2}{7} a^{5} + \frac{1}{7} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{49} a^{14} + \frac{2}{49} a^{13} - \frac{1}{49} a^{12} + \frac{4}{49} a^{11} + \frac{18}{49} a^{10} + \frac{18}{49} a^{9} - \frac{15}{49} a^{8} - \frac{11}{49} a^{7} - \frac{20}{49} a^{6} - \frac{1}{7} a^{5} + \frac{3}{7} a^{4} + \frac{2}{7} a^{3} - \frac{11}{49} a^{2} + \frac{5}{49} a - \frac{12}{49}$, $\frac{1}{483905981189415911749} a^{15} + \frac{26457083853594853}{5437145856060852941} a^{14} - \frac{3572483867063327812}{69129425884202273107} a^{13} + \frac{16949511065086199804}{69129425884202273107} a^{12} + \frac{82017640681867249}{28465057717024465397} a^{11} - \frac{118335188752759122929}{483905981189415911749} a^{10} + \frac{173017440585988759629}{483905981189415911749} a^{9} - \frac{9716646182495036198}{483905981189415911749} a^{8} - \frac{192855717058405506701}{483905981189415911749} a^{7} + \frac{2779561761197332908}{28465057717024465397} a^{6} + \frac{26188806621746750372}{69129425884202273107} a^{5} - \frac{586055948891079111}{9875632269171753301} a^{4} - \frac{137237226512466530447}{483905981189415911749} a^{3} - \frac{191447153305758966046}{483905981189415911749} a^{2} + \frac{152167897571805166416}{483905981189415911749} a - \frac{6993786818755609448}{28465057717024465397}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{12659250492361562}{580919545245397253} a^{15} - \frac{4493038252648755}{45690301311435739} a^{14} - \frac{167481261929052849}{4066436816717780771} a^{13} + \frac{2659709190111965772}{4066436816717780771} a^{12} + \frac{1617501937168090006}{4066436816717780771} a^{11} - \frac{14780991900868041249}{4066436816717780771} a^{10} - \frac{19416449909576990566}{4066436816717780771} a^{9} + \frac{64607634875738633361}{4066436816717780771} a^{8} + \frac{117420069748751163494}{4066436816717780771} a^{7} - \frac{115922129905132979712}{4066436816717780771} a^{6} - \frac{59021711575544727969}{580919545245397253} a^{5} - \frac{20526172641790502826}{580919545245397253} a^{4} + \frac{77789271972274543622}{580919545245397253} a^{3} + \frac{685759266485703480009}{4066436816717780771} a^{2} + \frac{324327095702025334668}{4066436816717780771} a + \frac{54585790554706814199}{4066436816717780771} \) (order $8$)
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:  \( 320264.414107 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T799:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n799 are not computed
Character table for t16n799 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.0.1088.2, 4.4.4352.1, \(\Q(\zeta_{8})\), 8.0.18939904.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.24.22$x^{8} + 4 x^{4} + 36$$8$$1$$24$$D_4\times C_2$$[2, 3, 4]^{2}$
2.8.20.55$x^{8} + 4 x^{6} + 4 x^{5} + 6 x^{4} + 2$$8$$1$$20$$Q_8:C_2$$[2, 3, 3]^{2}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
89Data not computed