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Properties

Label 16.0.60516574977...0000.8
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 7^{8}$
Root discriminant $30.65$
Ramified primes $2, 3, 5, 7$
Class number $32$ (GRH)
Class group $[4, 8]$ (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![2101, -9304, 3134, 11566, 30374, 3214, -5457, -7282, 4686, 2200, -1452, -442, 254, 50, -25, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 25*x^14 + 50*x^13 + 254*x^12 - 442*x^11 - 1452*x^10 + 2200*x^9 + 4686*x^8 - 7282*x^7 - 5457*x^6 + 3214*x^5 + 30374*x^4 + 11566*x^3 + 3134*x^2 - 9304*x + 2101)
 
gp: K = bnfinit(x^16 - 2*x^15 - 25*x^14 + 50*x^13 + 254*x^12 - 442*x^11 - 1452*x^10 + 2200*x^9 + 4686*x^8 - 7282*x^7 - 5457*x^6 + 3214*x^5 + 30374*x^4 + 11566*x^3 + 3134*x^2 - 9304*x + 2101, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 25 x^{14} + 50 x^{13} + 254 x^{12} - 442 x^{11} - 1452 x^{10} + 2200 x^{9} + 4686 x^{8} - 7282 x^{7} - 5457 x^{6} + 3214 x^{5} + 30374 x^{4} + 11566 x^{3} + 3134 x^{2} - 9304 x + 2101 \)

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:  \(605165749776000000000000=2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.65$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(420=2^{2}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{420}(1,·)$, $\chi_{420}(323,·)$, $\chi_{420}(71,·)$, $\chi_{420}(13,·)$, $\chi_{420}(337,·)$, $\chi_{420}(83,·)$, $\chi_{420}(407,·)$, $\chi_{420}(349,·)$, $\chi_{420}(97,·)$, $\chi_{420}(419,·)$, $\chi_{420}(167,·)$, $\chi_{420}(169,·)$, $\chi_{420}(239,·)$, $\chi_{420}(181,·)$, $\chi_{420}(251,·)$, $\chi_{420}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{44} a^{11} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{21}{44} a - \frac{1}{2}$, $\frac{1}{176} a^{12} - \frac{1}{88} a^{11} - \frac{1}{16} a^{10} + \frac{1}{8} a^{9} - \frac{7}{16} a^{8} + \frac{1}{4} a^{7} - \frac{5}{16} a^{6} - \frac{1}{8} a^{5} - \frac{5}{16} a^{4} - \frac{3}{8} a^{3} - \frac{3}{44} a^{2} + \frac{3}{22} a - \frac{1}{16}$, $\frac{1}{176} a^{13} + \frac{1}{176} a^{11} - \frac{3}{16} a^{9} + \frac{3}{8} a^{8} + \frac{3}{16} a^{7} + \frac{1}{4} a^{6} + \frac{7}{16} a^{5} + \frac{2}{11} a^{3} + \frac{21}{176} a - \frac{1}{8}$, $\frac{1}{296384} a^{14} - \frac{127}{148192} a^{13} - \frac{33}{13472} a^{12} + \frac{43}{9262} a^{11} - \frac{331}{6736} a^{10} - \frac{5719}{13472} a^{9} - \frac{77}{1684} a^{8} - \frac{1297}{13472} a^{7} + \frac{4193}{13472} a^{6} + \frac{483}{6736} a^{5} + \frac{75393}{296384} a^{4} + \frac{36471}{148192} a^{3} - \frac{2373}{26944} a^{2} + \frac{18675}{74096} a + \frac{4195}{26944}$, $\frac{1}{788492567314437646144} a^{15} - \frac{591698762848327}{394246283657218823072} a^{14} + \frac{95375504142368965}{35840571241565347552} a^{13} - \frac{11267591109476928}{12320196364288088221} a^{12} + \frac{196568976504518093}{49280785457152352884} a^{11} + \frac{3500636315169257337}{35840571241565347552} a^{10} + \frac{1940214473176756729}{17920285620782673776} a^{9} - \frac{16635714564439565061}{35840571241565347552} a^{8} + \frac{17881195065347656079}{35840571241565347552} a^{7} - \frac{6746359470061995569}{17920285620782673776} a^{6} - \frac{234607360371365994123}{788492567314437646144} a^{5} - \frac{29778959943201336481}{394246283657218823072} a^{4} - \frac{22705621665860136917}{71681142483130695104} a^{3} + \frac{73423562696307104575}{197123141828609411536} a^{2} - \frac{121751094923296112443}{788492567314437646144} a + \frac{3120829212465688705}{8960142810391336888}$

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

Class group and class number

$C_{4}\times C_{8}$, which has order $32$ (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:  \( -\frac{135658817260507}{620371807485788864} a^{15} + \frac{30976112772923}{310185903742894432} a^{14} + \frac{1887848240981345}{310185903742894432} a^{13} - \frac{88577734674347}{38773237967861804} a^{12} - \frac{11109339251179907}{155092951871447216} a^{11} + \frac{175624209166849}{28198718522081312} a^{10} + \frac{1646174414149053}{3524839815260164} a^{9} + \frac{1274354616739823}{28198718522081312} a^{8} - \frac{51323884261696179}{28198718522081312} a^{7} - \frac{1916306320323637}{14099359261040656} a^{6} + \frac{2530368078893884605}{620371807485788864} a^{5} + \frac{411212021244243957}{310185903742894432} a^{4} - \frac{5883956780913876059}{620371807485788864} a^{3} - \frac{847998471012210605}{77546475935723608} a^{2} - \frac{2947220539064520275}{620371807485788864} a + \frac{35617412783857187}{14099359261040656} \) (order $10$)
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:  \( 76884.6053994 \) (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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{3}, \sqrt{-35})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{15}, \sqrt{-21})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{5}, \sqrt{-21})\), \(\Q(\sqrt{-7}, \sqrt{15})\), \(\Q(\zeta_{5})\), 4.4.6125.1, 4.0.18000.1, 4.4.882000.1, 8.0.31116960000.6, 8.0.37515625.1, 8.0.777924000000.4, 8.0.324000000.3, 8.8.777924000000.1, 8.0.777924000000.10, 8.0.777924000000.2

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 R R ${\href{/LocalNumberField/11.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$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}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$