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Properties

Label 16.0.14473837877...5376.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 97^{8}$
Root discriminant $136.47$
Ramified primes $2, 3, 97$
Class number $1740800$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 10, 2720]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![270839247073, -1387724424, 67451890556, -59931840, 7647922624, 4129104, 518313508, 294912, 23084937, 4368, 696224, -24, 13978, 0, 172, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 172*x^14 + 13978*x^12 - 24*x^11 + 696224*x^10 + 4368*x^9 + 23084937*x^8 + 294912*x^7 + 518313508*x^6 + 4129104*x^5 + 7647922624*x^4 - 59931840*x^3 + 67451890556*x^2 - 1387724424*x + 270839247073)
 
gp: K = bnfinit(x^16 + 172*x^14 + 13978*x^12 - 24*x^11 + 696224*x^10 + 4368*x^9 + 23084937*x^8 + 294912*x^7 + 518313508*x^6 + 4129104*x^5 + 7647922624*x^4 - 59931840*x^3 + 67451890556*x^2 - 1387724424*x + 270839247073, 1)
 

Normalized defining polynomial

\( x^{16} + 172 x^{14} + 13978 x^{12} - 24 x^{11} + 696224 x^{10} + 4368 x^{9} + 23084937 x^{8} + 294912 x^{7} + 518313508 x^{6} + 4129104 x^{5} + 7647922624 x^{4} - 59931840 x^{3} + 67451890556 x^{2} - 1387724424 x + 270839247073 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14473837877661054916170965242085376=2^{48}\cdot 3^{8}\cdot 97^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $136.47$
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:  $2, 3, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4656=2^{4}\cdot 3\cdot 97\)
Dirichlet character group:    $\lbrace$$\chi_{4656}(1,·)$, $\chi_{4656}(581,·)$, $\chi_{4656}(775,·)$, $\chi_{4656}(971,·)$, $\chi_{4656}(1165,·)$, $\chi_{4656}(1745,·)$, $\chi_{4656}(1939,·)$, $\chi_{4656}(2135,·)$, $\chi_{4656}(2329,·)$, $\chi_{4656}(2909,·)$, $\chi_{4656}(3103,·)$, $\chi_{4656}(3299,·)$, $\chi_{4656}(3493,·)$, $\chi_{4656}(4073,·)$, $\chi_{4656}(4267,·)$, $\chi_{4656}(4463,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4883168447} a^{14} + \frac{1272738164}{4883168447} a^{13} - \frac{259742802}{4883168447} a^{12} - \frac{1091496515}{4883168447} a^{11} - \frac{909080740}{4883168447} a^{10} - \frac{383246484}{4883168447} a^{9} - \frac{1907772988}{4883168447} a^{8} + \frac{1372804213}{4883168447} a^{7} - \frac{1978886815}{4883168447} a^{6} + \frac{430735899}{4883168447} a^{5} + \frac{939579253}{4883168447} a^{4} + \frac{1327646123}{4883168447} a^{3} + \frac{390433603}{4883168447} a^{2} - \frac{52300998}{4883168447} a - \frac{954632851}{4883168447}$, $\frac{1}{411923991392898753914612488144359917835637485263} a^{15} + \frac{32593826289266396968815864234280674772}{411923991392898753914612488144359917835637485263} a^{14} + \frac{102200765351977073374008935464534504273857021618}{411923991392898753914612488144359917835637485263} a^{13} + \frac{106451901498894977392896890222144244519317062912}{411923991392898753914612488144359917835637485263} a^{12} - \frac{7882897118727631753187638905094328533029743465}{411923991392898753914612488144359917835637485263} a^{11} - \frac{104123496421515899330789270304820172176531978387}{411923991392898753914612488144359917835637485263} a^{10} + \frac{169036578046763662124932353327695969199668598301}{411923991392898753914612488144359917835637485263} a^{9} - \frac{83054451851298627648123669330879522464410915447}{411923991392898753914612488144359917835637485263} a^{8} + \frac{100305628843188728033462238180073243875365831411}{411923991392898753914612488144359917835637485263} a^{7} - \frac{2586746014881153951915755802569776577685106346}{24230823023111691406741911067315289284449263839} a^{6} - \frac{124100988044160789950879395580579814660304357744}{411923991392898753914612488144359917835637485263} a^{5} - \frac{34702728673021441724217871114370640911470791193}{411923991392898753914612488144359917835637485263} a^{4} - \frac{7279171954405782440651817346309383976054577571}{24230823023111691406741911067315289284449263839} a^{3} - \frac{54270335188628852559333749109123906265229259283}{411923991392898753914612488144359917835637485263} a^{2} + \frac{11214897170181593425906850162992549591573815249}{411923991392898753914612488144359917835637485263} a + \frac{182858435040361171221815199896951436655382073}{1723531344740162150270345138679330200149110817}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{10}\times C_{2720}$, which has order $1740800$ (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:  $7$
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:  \( 11964.310642723332 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-582}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-194}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-291}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-97}) \), \(\Q(\sqrt{3}, \sqrt{-194})\), \(\Q(\sqrt{2}, \sqrt{-291})\), \(\Q(\sqrt{6}, \sqrt{-97})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-97})\), \(\Q(\sqrt{2}, \sqrt{-97})\), \(\Q(\sqrt{6}, \sqrt{-194})\), 4.0.19269632.5, 4.4.18432.1, 4.0.173426688.5, \(\Q(\zeta_{16})^+\), 8.0.469950251728896.68, 8.0.30076816110649344.68, 8.0.30076816110649344.137, 8.0.120307264442597376.5, \(\Q(\zeta_{48})^+\), 8.0.1485274869661696.25, 8.0.120307264442597376.4

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$