↑

Properties

Label 16.8.68372792983...489.15
Degree $16$
Signature $[8, 4]$
Discriminant $17^{14}\cdot 67^{8}$
Root discriminant $97.65$
Ramified primes $17, 67$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9916201, -19306519, 218575, 10060145, -13031290, 6911503, 2452222, -1272579, 671213, -110764, -583, 4318, -2055, 215, -20, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 20*x^14 + 215*x^13 - 2055*x^12 + 4318*x^11 - 583*x^10 - 110764*x^9 + 671213*x^8 - 1272579*x^7 + 2452222*x^6 + 6911503*x^5 - 13031290*x^4 + 10060145*x^3 + 218575*x^2 - 19306519*x + 9916201)
 
gp: K = bnfinit(x^16 - 4*x^15 - 20*x^14 + 215*x^13 - 2055*x^12 + 4318*x^11 - 583*x^10 - 110764*x^9 + 671213*x^8 - 1272579*x^7 + 2452222*x^6 + 6911503*x^5 - 13031290*x^4 + 10060145*x^3 + 218575*x^2 - 19306519*x + 9916201, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 20 x^{14} + 215 x^{13} - 2055 x^{12} + 4318 x^{11} - 583 x^{10} - 110764 x^{9} + 671213 x^{8} - 1272579 x^{7} + 2452222 x^{6} + 6911503 x^{5} - 13031290 x^{4} + 10060145 x^{3} + 218575 x^{2} - 19306519 x + 9916201 \)

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:  \(68372792983010839504523425519489=17^{14}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.65$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{4963032000314970473063813652053497218402963530223357955680286} a^{15} - \frac{251912512737062798879485040441987743850809931677677970761947}{4963032000314970473063813652053497218402963530223357955680286} a^{14} - \frac{1083908001763002259602028438568647581568608847271287099224251}{4963032000314970473063813652053497218402963530223357955680286} a^{13} + \frac{366287288005145651155585491631666027738266207100725354580772}{2481516000157485236531906826026748609201481765111678977840143} a^{12} + \frac{301548189015101725909088334191827989656025731992686866170183}{4963032000314970473063813652053497218402963530223357955680286} a^{11} + \frac{7041277519419462226014304118016884579060618678847189425295}{33308939599429332033985326523848974620154117652505758091814} a^{10} + \frac{1179218278886361186517206796085903721800699272182530913041477}{4963032000314970473063813652053497218402963530223357955680286} a^{9} + \frac{695302673848708117087584501475378743362356224257977646404607}{4963032000314970473063813652053497218402963530223357955680286} a^{8} + \frac{517002170122409442756293468652382210287978268363797264231811}{2481516000157485236531906826026748609201481765111678977840143} a^{7} + \frac{2225876615282599295720291635678722588074828293099819141962195}{4963032000314970473063813652053497218402963530223357955680286} a^{6} - \frac{1469939269025971353924312672183978822778514195096590273864791}{4963032000314970473063813652053497218402963530223357955680286} a^{5} - \frac{294799916139377136374530050915619430747116910311732962639207}{4963032000314970473063813652053497218402963530223357955680286} a^{4} + \frac{142057219300716291015472533982865082979791439691586889371703}{381771692331920805620293357850269016800227963863335227360022} a^{3} - \frac{257277534875519731286270189784038097112184550089648243132259}{2481516000157485236531906826026748609201481765111678977840143} a^{2} + \frac{224257652843488730344654499204417363790649977002017279837275}{4963032000314970473063813652053497218402963530223357955680286} a + \frac{134224951677922643535414679266790993283934060223616574965}{788033026407585022715753199754445414163696972090085416907}$

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:  $11$
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:  \( 1640914604.05 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

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 $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.4.1842010303097.2, 8.6.123414690307499.1, 8.6.7259687665147.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
67Data not computed