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Properties

Label 16.0.10031792466...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 5^{12}\cdot 41^{4}\cdot 115001^{2}$
Root discriminant $205.39$
Ramified primes $2, 5, 41, 115001$
Class number $16$ (GRH)
Class group $[2, 8]$ (GRH)
Galois group 16T1605

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1396914918855625, 0, -101932286360000, 0, 3199905061000, 0, -60269405000, 0, 779073750, 0, -7400800, 0, 56040, 0, -280, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 280*x^14 + 56040*x^12 - 7400800*x^10 + 779073750*x^8 - 60269405000*x^6 + 3199905061000*x^4 - 101932286360000*x^2 + 1396914918855625)
 
gp: K = bnfinit(x^16 - 280*x^14 + 56040*x^12 - 7400800*x^10 + 779073750*x^8 - 60269405000*x^6 + 3199905061000*x^4 - 101932286360000*x^2 + 1396914918855625, 1)
 

Normalized defining polynomial

\( x^{16} - 280 x^{14} + 56040 x^{12} - 7400800 x^{10} + 779073750 x^{8} - 60269405000 x^{6} + 3199905061000 x^{4} - 101932286360000 x^{2} + 1396914918855625 \)

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:  \(10031792466827489906262016000000000000=2^{40}\cdot 5^{12}\cdot 41^{4}\cdot 115001^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $205.39$
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:  $2, 5, 41, 115001$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{10} a^{4} - \frac{1}{2}$, $\frac{1}{10} a^{5} - \frac{1}{2} a$, $\frac{1}{10} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{400} a^{8} + \frac{1}{4} a^{2} + \frac{7}{16}$, $\frac{1}{400} a^{9} + \frac{1}{4} a^{3} + \frac{7}{16} a$, $\frac{1}{800} a^{10} - \frac{1}{800} a^{8} + \frac{1}{40} a^{4} - \frac{13}{32} a^{2} - \frac{7}{32}$, $\frac{1}{800} a^{11} - \frac{1}{800} a^{9} + \frac{1}{40} a^{5} - \frac{13}{32} a^{3} - \frac{7}{32} a$, $\frac{1}{12000} a^{12} + \frac{1}{2400} a^{8} + \frac{1}{120} a^{6} + \frac{23}{480} a^{4} + \frac{1}{24} a^{2} + \frac{23}{96}$, $\frac{1}{12000} a^{13} + \frac{1}{2400} a^{9} + \frac{1}{120} a^{7} + \frac{23}{480} a^{5} + \frac{1}{24} a^{3} + \frac{23}{96} a$, $\frac{1}{422547606179499623886146710269419544000} a^{14} + \frac{10175655407548528409628340963440799}{422547606179499623886146710269419544000} a^{12} + \frac{9941774899286413431233563607502683}{16901904247179984955445868410776781760} a^{10} - \frac{5604075402637289647998329541892227}{9389946803988880530803260228209323200} a^{8} - \frac{59092219760134741128508446859743653}{1877989360797776106160652045641864640} a^{6} + \frac{164583802098552728547031888458049073}{3380380849435996991089173682155356352} a^{4} + \frac{533839995086469888410777050031450519}{1126793616478665663696391227385118784} a^{2} - \frac{7307878899774723821936354762081}{29394360478917548465571374876352}$, $\frac{1}{5493118880333495110519907233502454072000} a^{15} + \frac{10175655407548528409628340963440799}{5493118880333495110519907233502454072000} a^{13} - \frac{584112534772817368673052247366615901}{1098623776066699022103981446700490814400} a^{11} + \frac{41345658617307113006017971599154389}{122069308451855446900442382966721201600} a^{9} + \frac{9900516639972528422119750592649447}{1877989360797776106160652045641864640} a^{7} + \frac{3358204647569761386052039703906762629}{219724755213339804420796289340098162880} a^{5} - \frac{3198663859499110122583518890681755453}{14648317014222653628053085956006544192} a^{3} - \frac{113862435635850837009632588688857}{382126686225928130052427873392576} a$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  \( 169085130823 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1605:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 124 conjugacy class representatives for t16n1605 are not computed
Character table for t16n1605 is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), 4.0.65600.5, 8.0.4303360000.6

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$5$5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
115001Data not computed