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Properties

Label 16.8.12515348857...9089.5
Degree $16$
Signature $[8, 4]$
Discriminant $61^{8}\cdot 97^{14}$
Root discriminant $427.65$
Ramified primes $61, 97$
Class number $32$ (GRH)
Class group $[4, 8]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22898045041, 0, -35054114934, 0, 7300822245, 0, -309207550, 0, -11466300, 0, -134374, 0, -2259, 0, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 18*x^14 - 2259*x^12 - 134374*x^10 - 11466300*x^8 - 309207550*x^6 + 7300822245*x^4 - 35054114934*x^2 + 22898045041)
 
gp: K = bnfinit(x^16 + 18*x^14 - 2259*x^12 - 134374*x^10 - 11466300*x^8 - 309207550*x^6 + 7300822245*x^4 - 35054114934*x^2 + 22898045041, 1)
 

Normalized defining polynomial

\( x^{16} + 18 x^{14} - 2259 x^{12} - 134374 x^{10} - 11466300 x^{8} - 309207550 x^{6} + 7300822245 x^{4} - 35054114934 x^{2} + 22898045041 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1251534885791204872804259841565797642149089=61^{8}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $427.65$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{3}{8} a^{2} - \frac{3}{8} a - \frac{3}{8}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{6} - \frac{1}{8} a^{3} + \frac{1}{8}$, $\frac{1}{264} a^{10} + \frac{5}{88} a^{8} - \frac{7}{88} a^{6} - \frac{19}{88} a^{4} + \frac{15}{88} a^{2} - \frac{1}{2} a + \frac{89}{264}$, $\frac{1}{528} a^{11} - \frac{1}{528} a^{10} + \frac{5}{176} a^{9} - \frac{5}{176} a^{8} - \frac{7}{176} a^{7} - \frac{15}{176} a^{6} + \frac{25}{176} a^{5} - \frac{25}{176} a^{4} + \frac{15}{176} a^{3} + \frac{73}{176} a^{2} + \frac{89}{528} a - \frac{23}{528}$, $\frac{1}{12672} a^{12} - \frac{1}{3168} a^{10} + \frac{13}{1056} a^{8} - \frac{1}{8} a^{7} - \frac{69}{704} a^{6} + \frac{127}{1056} a^{4} + \frac{221}{3168} a^{2} - \frac{3}{8} a - \frac{2615}{12672}$, $\frac{1}{9858816} a^{13} - \frac{1}{25344} a^{12} - \frac{1357}{2464704} a^{11} - \frac{1}{576} a^{10} - \frac{3203}{821568} a^{9} + \frac{59}{2112} a^{8} - \frac{3327}{49792} a^{7} + \frac{37}{1408} a^{6} + \frac{49651}{821568} a^{5} - \frac{427}{2112} a^{4} - \frac{98419}{2464704} a^{3} + \frac{427}{6336} a^{2} - \frac{1975895}{9858816} a - \frac{6409}{25344}$, $\frac{1}{27729040566032042558402138112} a^{14} + \frac{83698859320077136330663}{27729040566032042558402138112} a^{12} + \frac{514353322381348804150979}{866532517688501329950066816} a^{10} - \frac{268479098747415229021144961}{4621506761005340426400356352} a^{8} - \frac{53921558736994723493919775}{4621506761005340426400356352} a^{6} - \frac{7917330452350223708898467}{39387841713113696815912128} a^{4} - \frac{7743626320714043472276516955}{27729040566032042558402138112} a^{2} - \frac{53322178589699268216949}{183246479774995159683072}$, $\frac{1}{21573193560372929110436863451136} a^{15} - \frac{1}{55458081132064085116804276224} a^{14} + \frac{83698859320077136330663}{21573193560372929110436863451136} a^{13} - \frac{83698859320077136330663}{55458081132064085116804276224} a^{12} - \frac{380234783237717720416332925}{674162298761654034701151982848} a^{11} - \frac{514353322381348804150979}{1733065035377002659900133632} a^{10} + \frac{466760613230707111542548095}{3595532260062154851739477241856} a^{9} + \frac{268479098747415229021144961}{9243013522010680852800712704} a^{8} - \frac{124939638350449489427812640287}{3595532260062154851739477241856} a^{7} + \frac{53921558736994723493919775}{9243013522010680852800712704} a^{6} + \frac{82253192466288330732866420447}{337081149380827017350575991424} a^{5} + \frac{7917330452350223708898467}{78775683426227393631824256} a^{4} + \frac{920549027173949563085140515749}{21573193560372929110436863451136} a^{3} + \frac{7743626320714043472276516955}{55458081132064085116804276224} a^{2} - \frac{28063657196923618487348341}{142565761264946234233430016} a - \frac{129924301185295891466123}{366492959549990319366144}$

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

Class group and class number

$C_{4}\times C_{8}$, which has order $32$ (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:  $11$
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:  \( 368918538667000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{97}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{5917}) \), 4.4.3396056233.1, 4.4.912673.1, \(\Q(\sqrt{61}, \sqrt{97})\), 8.4.1118720199956720578033.1 x2, 8.4.300650416543058473.1 x2, 8.8.11533197937698150289.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
61Data not computed
$97$97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$