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Properties

Label 16.4.90390672967...1729.1
Degree $16$
Signature $[4, 6]$
Discriminant $61^{4}\cdot 97^{14}$
Root discriminant $153.02$
Ramified primes $61, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-84718247, 3898849, -66743118, -128866001, -50423708, -20436854, -5394505, -1208372, -56512, -5105, -2714, 5706, -93, 160, 38, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 38*x^14 + 160*x^13 - 93*x^12 + 5706*x^11 - 2714*x^10 - 5105*x^9 - 56512*x^8 - 1208372*x^7 - 5394505*x^6 - 20436854*x^5 - 50423708*x^4 - 128866001*x^3 - 66743118*x^2 + 3898849*x - 84718247)
 
gp: K = bnfinit(x^16 - 2*x^15 + 38*x^14 + 160*x^13 - 93*x^12 + 5706*x^11 - 2714*x^10 - 5105*x^9 - 56512*x^8 - 1208372*x^7 - 5394505*x^6 - 20436854*x^5 - 50423708*x^4 - 128866001*x^3 - 66743118*x^2 + 3898849*x - 84718247, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 38 x^{14} + 160 x^{13} - 93 x^{12} + 5706 x^{11} - 2714 x^{10} - 5105 x^{9} - 56512 x^{8} - 1208372 x^{7} - 5394505 x^{6} - 20436854 x^{5} - 50423708 x^{4} - 128866001 x^{3} - 66743118 x^{2} + 3898849 x - 84718247 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(90390672967514567934462041097091729=61^{4}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $153.02$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} + \frac{1}{3} a^{8} - \frac{1}{6} a^{6} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{10} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{18} a^{14} - \frac{1}{18} a^{13} - \frac{1}{18} a^{11} - \frac{1}{18} a^{10} - \frac{7}{18} a^{8} + \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{5}{18} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} + \frac{7}{18} a^{2} - \frac{7}{18} a + \frac{2}{9}$, $\frac{1}{60148379325040561004995851247636302651987163719208666565538} a^{15} + \frac{377760435646602012321068588569321875211514397564684279361}{20049459775013520334998617082545434217329054573069555521846} a^{14} + \frac{1470147793588917449694233773549475181842194752477505603693}{30074189662520280502497925623818151325993581859604333282769} a^{13} + \frac{2478650987167613892533989502537387019012730070870054833829}{60148379325040561004995851247636302651987163719208666565538} a^{12} + \frac{8252665504934495815306126808369423861621971677468364745359}{60148379325040561004995851247636302651987163719208666565538} a^{11} + \frac{1699356631075515157224925473277063025736201981454358194084}{30074189662520280502497925623818151325993581859604333282769} a^{10} + \frac{8662707816296535792625075758615893237340466049505743148107}{60148379325040561004995851247636302651987163719208666565538} a^{9} - \frac{16704454599942464572397173632925139697069676639086997393513}{60148379325040561004995851247636302651987163719208666565538} a^{8} - \frac{229381309937468582491968922865562573578335816869458382784}{3341576629168920055833102847090905702888175762178259253641} a^{7} + \frac{26017880245115871908341279718917522056369384188455987539025}{60148379325040561004995851247636302651987163719208666565538} a^{6} + \frac{12197661633896415511906059802326519939802646041316436966985}{60148379325040561004995851247636302651987163719208666565538} a^{5} - \frac{4210079627680232639939365018274443794156273600953145941701}{10024729887506760167499308541272717108664527286534777760923} a^{4} + \frac{22612357317035513184654433228414043315947078058557430133541}{60148379325040561004995851247636302651987163719208666565538} a^{3} - \frac{5363911901505614827268955006605369312496551962218158971057}{20049459775013520334998617082545434217329054573069555521846} a^{2} - \frac{2018821512107696518034986597553640052419050243907321669142}{10024729887506760167499308541272717108664527286534777760923} a + \frac{145709590303535565064638835526770411104022741396442470}{20899367381876497916954778056857645118828062445868195471}$

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:  $9$
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:  \( 74375960429.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

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 $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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}$