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Properties

Label 16.12.1213988839...1633.1
Degree $16$
Signature $[12, 2]$
Discriminant $61^{8}\cdot 97^{15}$
Root discriminant $569.20$
Ramified primes $61, 97$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $(C_2\times OD_{16}).D_4$ (as 16T591)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-18891513523, -511640977, 19605075113, 1795154696, -6692210494, -924037993, 905586480, 162030550, -48291565, -8926458, 576156, 180490, 14110, -518, -235, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 235*x^14 - 518*x^13 + 14110*x^12 + 180490*x^11 + 576156*x^10 - 8926458*x^9 - 48291565*x^8 + 162030550*x^7 + 905586480*x^6 - 924037993*x^5 - 6692210494*x^4 + 1795154696*x^3 + 19605075113*x^2 - 511640977*x - 18891513523)
 
gp: K = bnfinit(x^16 - 4*x^15 - 235*x^14 - 518*x^13 + 14110*x^12 + 180490*x^11 + 576156*x^10 - 8926458*x^9 - 48291565*x^8 + 162030550*x^7 + 905586480*x^6 - 924037993*x^5 - 6692210494*x^4 + 1795154696*x^3 + 19605075113*x^2 - 511640977*x - 18891513523, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 235 x^{14} - 518 x^{13} + 14110 x^{12} + 180490 x^{11} + 576156 x^{10} - 8926458 x^{9} - 48291565 x^{8} + 162030550 x^{7} + 905586480 x^{6} - 924037993 x^{5} - 6692210494 x^{4} + 1795154696 x^{3} + 19605075113 x^{2} - 511640977 x - 18891513523 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(121398883921746872662013204631882371288461633=61^{8}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $569.20$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{811} a^{14} - \frac{43}{811} a^{13} + \frac{381}{811} a^{12} + \frac{239}{811} a^{11} + \frac{371}{811} a^{10} + \frac{30}{811} a^{9} - \frac{309}{811} a^{8} - \frac{96}{811} a^{7} + \frac{135}{811} a^{6} + \frac{131}{811} a^{5} - \frac{381}{811} a^{4} - \frac{51}{811} a^{3} + \frac{36}{811} a^{2} - \frac{300}{811} a + \frac{308}{811}$, $\frac{1}{51977817079696877892693713301339089159696684212734717276998302039227917} a^{15} + \frac{16635247644691504646400057385838307301556821407202603046977928891748}{51977817079696877892693713301339089159696684212734717276998302039227917} a^{14} - \frac{12917669025915286192565034142857687641281723881879510884594740283063838}{51977817079696877892693713301339089159696684212734717276998302039227917} a^{13} - \frac{6619570954426066965600358755432613438909828766686045401666153425740024}{51977817079696877892693713301339089159696684212734717276998302039227917} a^{12} + \frac{23629010822688670753110420328779886966793820495128259404094201955122413}{51977817079696877892693713301339089159696684212734717276998302039227917} a^{11} + \frac{24397596719024091609610427873227173450010727544092166351282559924129548}{51977817079696877892693713301339089159696684212734717276998302039227917} a^{10} - \frac{3146802563827540331982579519919341266420190384804844533878672315897846}{51977817079696877892693713301339089159696684212734717276998302039227917} a^{9} + \frac{16568623955944672802638243661699293669777228340303019149141190661086051}{51977817079696877892693713301339089159696684212734717276998302039227917} a^{8} - \frac{6896464813912709879503008358127623965995422296944419089836305584108574}{51977817079696877892693713301339089159696684212734717276998302039227917} a^{7} - \frac{16961412360247124360153123291364124572661590318508963810608229822501302}{51977817079696877892693713301339089159696684212734717276998302039227917} a^{6} - \frac{14227873013702559664044770518188125401213615631518911851588849893501221}{51977817079696877892693713301339089159696684212734717276998302039227917} a^{5} + \frac{9997885316925857628965661257892954078385732863037404324781256321577534}{51977817079696877892693713301339089159696684212734717276998302039227917} a^{4} + \frac{9503642621134623404580788254841667908007500252881185204415398696241538}{51977817079696877892693713301339089159696684212734717276998302039227917} a^{3} - \frac{15317591946273581607262644409988387880094361401684980271554233229445392}{51977817079696877892693713301339089159696684212734717276998302039227917} a^{2} - \frac{5478808574940934666007846854034192103873225635047623037114155542011787}{51977817079696877892693713301339089159696684212734717276998302039227917} a + \frac{16958073071079761791112094415483083702795875649556953966471991478887452}{51977817079696877892693713301339089159696684212734717276998302039227917}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $13$
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:  \( 3053510576340000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).D_4$ (as 16T591):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $(C_2\times OD_{16}).D_4$
Character table for $(C_2\times OD_{16}).D_4$ 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 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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