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Properties

Label 16.0.80320700706...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 5^{8}\cdot 43^{8}$
Root discriminant $98.64$
Ramified primes $2, 5, 43$
Class number $304500$ (GRH)
Class group $[5, 60900]$ (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![1140903199, -493914464, 592976072, -210112840, 135999792, -39901264, 18083476, -4442312, 1555101, -322272, 91312, -15856, 3674, -504, 92, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 92*x^14 - 504*x^13 + 3674*x^12 - 15856*x^11 + 91312*x^10 - 322272*x^9 + 1555101*x^8 - 4442312*x^7 + 18083476*x^6 - 39901264*x^5 + 135999792*x^4 - 210112840*x^3 + 592976072*x^2 - 493914464*x + 1140903199)
 
gp: K = bnfinit(x^16 - 8*x^15 + 92*x^14 - 504*x^13 + 3674*x^12 - 15856*x^11 + 91312*x^10 - 322272*x^9 + 1555101*x^8 - 4442312*x^7 + 18083476*x^6 - 39901264*x^5 + 135999792*x^4 - 210112840*x^3 + 592976072*x^2 - 493914464*x + 1140903199, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 92 x^{14} - 504 x^{13} + 3674 x^{12} - 15856 x^{11} + 91312 x^{10} - 322272 x^{9} + 1555101 x^{8} - 4442312 x^{7} + 18083476 x^{6} - 39901264 x^{5} + 135999792 x^{4} - 210112840 x^{3} + 592976072 x^{2} - 493914464 x + 1140903199 \)

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:  \(80320700706231066139033600000000=2^{44}\cdot 5^{8}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.64$
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:  $2, 5, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3440=2^{4}\cdot 5\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{3440}(1,·)$, $\chi_{3440}(3269,·)$, $\chi_{3440}(3009,·)$, $\chi_{3440}(1289,·)$, $\chi_{3440}(1549,·)$, $\chi_{3440}(429,·)$, $\chi_{3440}(2321,·)$, $\chi_{3440}(2581,·)$, $\chi_{3440}(601,·)$, $\chi_{3440}(861,·)$, $\chi_{3440}(2149,·)$, $\chi_{3440}(2409,·)$, $\chi_{3440}(3181,·)$, $\chi_{3440}(689,·)$, $\chi_{3440}(1461,·)$, $\chi_{3440}(1721,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{352253076121449589954} a^{14} - \frac{1}{50321868017349941422} a^{13} - \frac{39792934203486673465}{352253076121449589954} a^{12} + \frac{31315533580097622952}{176126538060724794977} a^{11} - \frac{1195350605115440084}{176126538060724794977} a^{10} - \frac{31569709205957550506}{176126538060724794977} a^{9} + \frac{42281299634622379301}{352253076121449589954} a^{8} - \frac{79053088834599987882}{176126538060724794977} a^{7} + \frac{437303270715654260}{176126538060724794977} a^{6} + \frac{71306387056980862274}{176126538060724794977} a^{5} - \frac{171758615033581482755}{352253076121449589954} a^{4} - \frac{9732703003388824577}{25160934008674970711} a^{3} + \frac{40591713228607858115}{352253076121449589954} a^{2} + \frac{106327689836314345883}{352253076121449589954} a + \frac{775157039879170775}{5681501227765315967}$, $\frac{1}{2884857232851043228884382466} a^{15} + \frac{4094857}{2884857232851043228884382466} a^{14} + \frac{41409354656796995876566179}{1442428616425521614442191233} a^{13} + \frac{9973012026853252622249288}{1442428616425521614442191233} a^{12} + \frac{452686690719775482753277795}{2884857232851043228884382466} a^{11} + \frac{72319925188537416340212603}{1442428616425521614442191233} a^{10} + \frac{64649335331969299014246966}{1442428616425521614442191233} a^{9} + \frac{287337693805191047530685939}{2884857232851043228884382466} a^{8} - \frac{962156310085515795730412243}{2884857232851043228884382466} a^{7} - \frac{283695554840514947928356018}{1442428616425521614442191233} a^{6} - \frac{472015788196529017135098224}{1442428616425521614442191233} a^{5} + \frac{1286469888292689694523996975}{2884857232851043228884382466} a^{4} - \frac{133839018239282668224454622}{1442428616425521614442191233} a^{3} - \frac{779189447933452929556714001}{2884857232851043228884382466} a^{2} + \frac{44086406956905041545513156}{1442428616425521614442191233} a - \frac{318649346329331804624851}{93059910737130426738205886}$

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

Class group and class number

$C_{5}\times C_{60900}$, which has order $304500$ (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:  \( 12198.951274811623 \) (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

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-215}) \), \(\Q(\sqrt{-86}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-430}) \), \(\Q(\sqrt{-43}) \), \(\Q(\sqrt{10}, \sqrt{-86})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{10}, \sqrt{-43})\), \(\Q(\sqrt{2}, \sqrt{-215})\), \(\Q(\sqrt{5}, \sqrt{-43})\), \(\Q(\sqrt{2}, \sqrt{-43})\), \(\Q(\sqrt{5}, \sqrt{-86})\), 4.4.51200.1, \(\Q(\zeta_{16})^+\), 4.0.3786752.2, 4.0.94668800.2, 8.0.8752130560000.17, 8.8.2621440000.1, 8.0.8962181693440000.84, 8.0.8962181693440000.42, 8.0.8962181693440000.81, 8.0.8962181693440000.49, 8.0.14339490709504.11

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$43$43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$