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Properties

Label 16.0.18346075763...5504.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{34}\cdot 17^{8}\cdot 11525081^{3}$
Root discriminant $379.29$
Ramified primes $2, 17, 11525081$
Class number $9088$ (GRH)
Class group $[2, 8, 568]$ (GRH)
Galois group 16T1472

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2453609642404, 892215487056, 703649768052, 171022222928, 78179443413, 13182702892, 4525440942, 502233900, 152675079, 9423992, 3150376, 73336, 39911, -196, 302, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 302*x^14 - 196*x^13 + 39911*x^12 + 73336*x^11 + 3150376*x^10 + 9423992*x^9 + 152675079*x^8 + 502233900*x^7 + 4525440942*x^6 + 13182702892*x^5 + 78179443413*x^4 + 171022222928*x^3 + 703649768052*x^2 + 892215487056*x + 2453609642404)
 
gp: K = bnfinit(x^16 - 4*x^15 + 302*x^14 - 196*x^13 + 39911*x^12 + 73336*x^11 + 3150376*x^10 + 9423992*x^9 + 152675079*x^8 + 502233900*x^7 + 4525440942*x^6 + 13182702892*x^5 + 78179443413*x^4 + 171022222928*x^3 + 703649768052*x^2 + 892215487056*x + 2453609642404, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 302 x^{14} - 196 x^{13} + 39911 x^{12} + 73336 x^{11} + 3150376 x^{10} + 9423992 x^{9} + 152675079 x^{8} + 502233900 x^{7} + 4525440942 x^{6} + 13182702892 x^{5} + 78179443413 x^{4} + 171022222928 x^{3} + 703649768052 x^{2} + 892215487056 x + 2453609642404 \)

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:  \(183460757637086369287254273534720689045504=2^{34}\cdot 17^{8}\cdot 11525081^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $379.29$
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:  $2, 17, 11525081$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{7} + \frac{1}{16} a^{6} - \frac{1}{2} a^{5} + \frac{7}{16} a^{4} + \frac{1}{8} a^{3} + \frac{1}{16} a^{2} + \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{16} a^{9} + \frac{1}{16} a^{7} - \frac{1}{8} a^{6} + \frac{7}{16} a^{5} + \frac{1}{16} a^{3} + \frac{1}{8} a^{2} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{16} a^{10} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} + \frac{1}{16} a^{2} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{96} a^{11} - \frac{1}{32} a^{10} - \frac{1}{32} a^{9} + \frac{1}{96} a^{8} - \frac{11}{96} a^{7} + \frac{1}{96} a^{6} + \frac{5}{96} a^{5} + \frac{41}{96} a^{4} - \frac{7}{24} a^{3} - \frac{11}{24} a^{2} - \frac{5}{24} a - \frac{1}{24}$, $\frac{1}{96} a^{12} - \frac{1}{48} a^{9} - \frac{1}{48} a^{8} + \frac{5}{48} a^{7} - \frac{11}{48} a^{6} - \frac{23}{48} a^{5} - \frac{31}{96} a^{4} + \frac{17}{48} a^{3} - \frac{13}{48} a^{2} + \frac{5}{24} a$, $\frac{1}{192} a^{13} - \frac{1}{192} a^{12} + \frac{1}{48} a^{10} + \frac{1}{12} a^{7} + \frac{1}{8} a^{6} - \frac{11}{64} a^{5} - \frac{55}{192} a^{4} - \frac{1}{16} a^{3} - \frac{1}{6} a^{2} - \frac{17}{48} a - \frac{3}{16}$, $\frac{1}{576} a^{14} + \frac{1}{576} a^{12} + \frac{1}{36} a^{10} + \frac{1}{72} a^{9} - \frac{1}{144} a^{8} - \frac{1}{9} a^{7} - \frac{23}{192} a^{6} + \frac{35}{72} a^{5} + \frac{7}{576} a^{4} - \frac{5}{36} a^{3} + \frac{1}{48} a^{2} + \frac{5}{18} a + \frac{13}{144}$, $\frac{1}{706514875078120522962458927813893857393746704225360204047764385694848} a^{15} + \frac{322073196057611094734143915977008140305744814451952191198556266689}{706514875078120522962458927813893857393746704225360204047764385694848} a^{14} - \frac{988741118698849777448039497990291991276497881545968164928512262265}{706514875078120522962458927813893857393746704225360204047764385694848} a^{13} + \frac{787239707236012225949978809092877998483688513047850771921381507}{1224462521799169017265960013542277049209266385139272450689366353024} a^{12} + \frac{1336265870348806974555710202101628612338881798019508593108961678707}{353257437539060261481229463906946928696873352112680102023882192847424} a^{11} - \frac{689994292661420165243172666967476005911544138687204615893737441733}{39250826393228917942358829322994103188541483568075566891542465871936} a^{10} + \frac{1594211296064582297313255206596296480526947620282409372924399014783}{353257437539060261481229463906946928696873352112680102023882192847424} a^{9} - \frac{10930087460917842246845529440322764430531559021223348834799254187229}{353257437539060261481229463906946928696873352112680102023882192847424} a^{8} - \frac{77694221376835967193523544778990682255192746379026501504577221469363}{706514875078120522962458927813893857393746704225360204047764385694848} a^{7} - \frac{136722329931613017498270206715656458021621145471528717237186165253803}{706514875078120522962458927813893857393746704225360204047764385694848} a^{6} + \frac{83912509236364935106696973338432233070553254136920242350453391759755}{706514875078120522962458927813893857393746704225360204047764385694848} a^{5} - \frac{152640773830039880571934697554877780566799702762881023608192498871097}{706514875078120522962458927813893857393746704225360204047764385694848} a^{4} - \frac{7131350402645607373393401966433185649597009568320013031278668654027}{22078589846191266342576841494184183043554584507042506376492637052964} a^{3} - \frac{14684467904897895448254154173250014201246035718867303410919129300109}{88314359384765065370307365976736732174218338028170025505970548211856} a^{2} - \frac{55304616785999552103051981574378255054601856120766285880860670394253}{176628718769530130740614731953473464348436676056340051011941096423712} a + \frac{14235227372541181789000720886569403683563564621927731024762051124427}{176628718769530130740614731953473464348436676056340051011941096423712}$

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

Class group and class number

$C_{2}\times C_{8}\times C_{568}$, which has order $9088$ (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:  \( 266041076739 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1472:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1472 are not computed
Character table for t16n1472 is not computed

Intermediate fields

\(\Q(\sqrt{34}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{2}, \sqrt{17})\), 8.0.15771013778653184.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.8.22.84$x^{8} + 4 x^{7} + 10 x^{4} + 4 x^{2} + 14$$8$$1$$22$$D_4$$[2, 3, 7/2]$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11525081Data not computed