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Properties

Label 16.8.14225707034...7073.1
Degree $16$
Signature $[8, 4]$
Discriminant $17^{15}\cdot 89^{6}$
Root discriminant $76.66$
Ramified primes $17, 89$
Class number $2$ (GRH)
Class group $[2]$ (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![12813683, 120137644, 36211563, 6621394, -3090174, -10705746, 1488111, -1009787, 376993, 5257, 13565, 2750, -1461, 55, -35, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 35*x^14 + 55*x^13 - 1461*x^12 + 2750*x^11 + 13565*x^10 + 5257*x^9 + 376993*x^8 - 1009787*x^7 + 1488111*x^6 - 10705746*x^5 - 3090174*x^4 + 6621394*x^3 + 36211563*x^2 + 120137644*x + 12813683)
 
gp: K = bnfinit(x^16 - 4*x^15 - 35*x^14 + 55*x^13 - 1461*x^12 + 2750*x^11 + 13565*x^10 + 5257*x^9 + 376993*x^8 - 1009787*x^7 + 1488111*x^6 - 10705746*x^5 - 3090174*x^4 + 6621394*x^3 + 36211563*x^2 + 120137644*x + 12813683, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 35 x^{14} + 55 x^{13} - 1461 x^{12} + 2750 x^{11} + 13565 x^{10} + 5257 x^{9} + 376993 x^{8} - 1009787 x^{7} + 1488111 x^{6} - 10705746 x^{5} - 3090174 x^{4} + 6621394 x^{3} + 36211563 x^{2} + 120137644 x + 12813683 \)

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:  \(1422570703415873252968445947073=17^{15}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.66$
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:  $17, 89$
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}{177677985040423621638976100085345708519140086151398625231398746299} a^{15} - \frac{53616293084848152439738034088593545566901353583462524143411770364}{177677985040423621638976100085345708519140086151398625231398746299} a^{14} - \frac{28987922929830684853456583469713025631432794001360618679730829903}{177677985040423621638976100085345708519140086151398625231398746299} a^{13} - \frac{41039971077449980451696215825784808926154752570896960531547201719}{177677985040423621638976100085345708519140086151398625231398746299} a^{12} - \frac{64056059014584994527345844262448142177960133945894501260129831955}{177677985040423621638976100085345708519140086151398625231398746299} a^{11} - \frac{38635201448262749626867737177248602151889124029045579660574522237}{177677985040423621638976100085345708519140086151398625231398746299} a^{10} - \frac{76674224472231219295068417604295954295052133221347233398505771167}{177677985040423621638976100085345708519140086151398625231398746299} a^{9} - \frac{4058391959483289068833572478506588517573906625585726339238353392}{177677985040423621638976100085345708519140086151398625231398746299} a^{8} - \frac{63521383714914559192074184090231443533698625609811786040383581938}{177677985040423621638976100085345708519140086151398625231398746299} a^{7} + \frac{57622299458239456035493835741531918921409994170738972247449254737}{177677985040423621638976100085345708519140086151398625231398746299} a^{6} + \frac{16827411170309923719847082522748046408833573157532734414416634366}{177677985040423621638976100085345708519140086151398625231398746299} a^{5} + \frac{768749420851467384419608751967594390076214907535886438296391095}{2651910224483934651328001493811129977897613226140277988528339497} a^{4} - \frac{1063557555087521625569038584984829428997429770349423802373003522}{2651910224483934651328001493811129977897613226140277988528339497} a^{3} + \frac{432228374547900623847754081919551696764376533592347430278336465}{1759187970697263580583921783023224836823169169815827972588106399} a^{2} + \frac{42378649584439763622502183242879686041172585708580951946084496281}{177677985040423621638976100085345708519140086151398625231398746299} a + \frac{1071403880975658607640128727021852959740389732097467154983738340}{2651910224483934651328001493811129977897613226140277988528339497}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 551193373.441 \) (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

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
$89$89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$