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Properties

Label 16.8.16808696443...8353.1
Degree $16$
Signature $[8, 4]$
Discriminant $61^{12}\cdot 97^{15}$
Root discriminant $1590.73$
Ramified primes $61, 97$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1512274356748426639, 784199220962810898, -204922171861025793, 41084287044150071, -3983226416739629, -210196292452445, 69548060234944, -5536048330415, 439598737165, -32204712967, 838802231, -2431318, 251441, 34778, -2205, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 2205*x^14 + 34778*x^13 + 251441*x^12 - 2431318*x^11 + 838802231*x^10 - 32204712967*x^9 + 439598737165*x^8 - 5536048330415*x^7 + 69548060234944*x^6 - 210196292452445*x^5 - 3983226416739629*x^4 + 41084287044150071*x^3 - 204922171861025793*x^2 + 784199220962810898*x - 1512274356748426639)
 
gp: K = bnfinit(x^16 - 7*x^15 - 2205*x^14 + 34778*x^13 + 251441*x^12 - 2431318*x^11 + 838802231*x^10 - 32204712967*x^9 + 439598737165*x^8 - 5536048330415*x^7 + 69548060234944*x^6 - 210196292452445*x^5 - 3983226416739629*x^4 + 41084287044150071*x^3 - 204922171861025793*x^2 + 784199220962810898*x - 1512274356748426639, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 2205 x^{14} + 34778 x^{13} + 251441 x^{12} - 2431318 x^{11} + 838802231 x^{10} - 32204712967 x^{9} + 439598737165 x^{8} - 5536048330415 x^{7} + 69548060234944 x^{6} - 210196292452445 x^{5} - 3983226416739629 x^{4} + 41084287044150071 x^{3} - 204922171861025793 x^{2} + 784199220962810898 x - 1512274356748426639 \)

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:  \(1680869644357963641125481571233506843563004905118353=61^{12}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1590.73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{176521103511028866795877774743494672182101067469863996470319363201818207166293180250068691575609245121710323091563769612494100944864087} a^{15} - \frac{74678609674075679278957898897461669636475794083058196918307246814867733997029173573523296624435317957648904436344026871747161983997623}{176521103511028866795877774743494672182101067469863996470319363201818207166293180250068691575609245121710323091563769612494100944864087} a^{14} + \frac{84904997031428265490271266992593060825065262700600303525873890256504146543766021273338056636897690847044909482417789477681874957904367}{176521103511028866795877774743494672182101067469863996470319363201818207166293180250068691575609245121710323091563769612494100944864087} a^{13} + \frac{58749389682051332737297579873109170592220306481877338785391109870656603755255420091131599355910149751584752342165897455896387371815509}{176521103511028866795877774743494672182101067469863996470319363201818207166293180250068691575609245121710323091563769612494100944864087} a^{12} + \frac{23051687244832562304246534268596142588956114142378237129507231578110177801219585109643341695427759623220257153456666222348659997735080}{176521103511028866795877774743494672182101067469863996470319363201818207166293180250068691575609245121710323091563769612494100944864087} a^{11} - \frac{10241379003434592007273053208574925423655419691738737419611116803859194510382200921755448543011197594414995915054370364812273253701735}{176521103511028866795877774743494672182101067469863996470319363201818207166293180250068691575609245121710323091563769612494100944864087} a^{10} - \frac{47098050301004702975069464251262941085272582666829549610867026415817217057056457389031447249348362425857214085575277959302630809954272}{176521103511028866795877774743494672182101067469863996470319363201818207166293180250068691575609245121710323091563769612494100944864087} a^{9} - \frac{20778782911843836893205878257092643372498427104757392346420836872592196877238742656956548803026874607095711838081322283452810377065780}{176521103511028866795877774743494672182101067469863996470319363201818207166293180250068691575609245121710323091563769612494100944864087} a^{8} + \frac{43610471622294781151988007211675186014774836846180978659951995559943756485434037223457513512629657741093638137814188861140272676635774}{176521103511028866795877774743494672182101067469863996470319363201818207166293180250068691575609245121710323091563769612494100944864087} a^{7} + \frac{28127430396910652603481936839631972140806107121638350587921723141191640369069816381156412409722806107304160117381401599706702766552845}{176521103511028866795877774743494672182101067469863996470319363201818207166293180250068691575609245121710323091563769612494100944864087} a^{6} - \frac{28334627015050525886307231484948075788422976870241933566791959697065997581621812824620268590123523843034073320365084545384998585782291}{176521103511028866795877774743494672182101067469863996470319363201818207166293180250068691575609245121710323091563769612494100944864087} a^{5} + \frac{87726292931102005253017177285362947190561220256388288045004456005827352868054647550880771814121878705286337785743599859748006905053752}{176521103511028866795877774743494672182101067469863996470319363201818207166293180250068691575609245121710323091563769612494100944864087} a^{4} + \frac{84323460881668832767959255683111314971617267633770801262374345863097490982944711940302115233245987623948926355496307732581662239725941}{176521103511028866795877774743494672182101067469863996470319363201818207166293180250068691575609245121710323091563769612494100944864087} a^{3} - \frac{35939660725132481210139351483636585059327649749130106864843516336613846053852637528286180433364953402248121453078358423542566206200823}{176521103511028866795877774743494672182101067469863996470319363201818207166293180250068691575609245121710323091563769612494100944864087} a^{2} + \frac{16626041787489257651957183868949548767516119616942232192954234768966587440688337938983650810721985983837542841618690981288155890419931}{176521103511028866795877774743494672182101067469863996470319363201818207166293180250068691575609245121710323091563769612494100944864087} a - \frac{22863426467266289224681461093411460922276973409281031652721207331035845769776863969321674095452274926745914329755566237079890820009416}{176521103511028866795877774743494672182101067469863996470319363201818207166293180250068691575609245121710323091563769612494100944864087}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, 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:  $11$
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:  \( 2104142378050000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.1118720199956720578033.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
97Data not computed