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Properties

Label 16.0.68455253549...0000.8
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 5^{8}\cdot 53^{8}$
Root discriminant $130.23$
Ramified primes $2, 5, 53$
Class number $2163168$ (GRH)
Class group $[6, 360528]$ (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![45036203521, -17113258880, 19910692468, -5873105256, 3650673056, -909551832, 393089516, -81567920, 26143449, -4020968, 993728, -107064, 20938, -1456, 228, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 228*x^14 - 1456*x^13 + 20938*x^12 - 107064*x^11 + 993728*x^10 - 4020968*x^9 + 26143449*x^8 - 81567920*x^7 + 393089516*x^6 - 909551832*x^5 + 3650673056*x^4 - 5873105256*x^3 + 19910692468*x^2 - 17113258880*x + 45036203521)
 
gp: K = bnfinit(x^16 - 8*x^15 + 228*x^14 - 1456*x^13 + 20938*x^12 - 107064*x^11 + 993728*x^10 - 4020968*x^9 + 26143449*x^8 - 81567920*x^7 + 393089516*x^6 - 909551832*x^5 + 3650673056*x^4 - 5873105256*x^3 + 19910692468*x^2 - 17113258880*x + 45036203521, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 228 x^{14} - 1456 x^{13} + 20938 x^{12} - 107064 x^{11} + 993728 x^{10} - 4020968 x^{9} + 26143449 x^{8} - 81567920 x^{7} + 393089516 x^{6} - 909551832 x^{5} + 3650673056 x^{4} - 5873105256 x^{3} + 19910692468 x^{2} - 17113258880 x + 45036203521 \)

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:  \(6845525354902535215356313600000000=2^{48}\cdot 5^{8}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $130.23$
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:  $2, 5, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4240=2^{4}\cdot 5\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{4240}(1,·)$, $\chi_{4240}(2119,·)$, $\chi_{4240}(2121,·)$, $\chi_{4240}(4239,·)$, $\chi_{4240}(849,·)$, $\chi_{4240}(211,·)$, $\chi_{4240}(2969,·)$, $\chi_{4240}(2331,·)$, $\chi_{4240}(1059,·)$, $\chi_{4240}(1061,·)$, $\chi_{4240}(3179,·)$, $\chi_{4240}(3181,·)$, $\chi_{4240}(1909,·)$, $\chi_{4240}(1271,·)$, $\chi_{4240}(4029,·)$, $\chi_{4240}(3391,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{27} a^{8} - \frac{4}{27} a^{7} - \frac{2}{27} a^{6} - \frac{7}{27} a^{5} - \frac{10}{27} a^{4} + \frac{1}{3} a^{3} - \frac{4}{9} a^{2} - \frac{2}{27} a - \frac{7}{27}$, $\frac{1}{27} a^{9} + \frac{1}{3} a^{7} + \frac{4}{9} a^{6} - \frac{11}{27} a^{5} - \frac{4}{27} a^{4} - \frac{1}{9} a^{3} + \frac{4}{27} a^{2} + \frac{4}{9} a - \frac{1}{27}$, $\frac{1}{10557} a^{10} - \frac{5}{10557} a^{9} + \frac{7}{459} a^{8} - \frac{614}{10557} a^{7} - \frac{1664}{3519} a^{6} - \frac{4010}{10557} a^{5} + \frac{244}{1173} a^{4} - \frac{1934}{10557} a^{3} - \frac{725}{10557} a^{2} - \frac{635}{10557} a - \frac{839}{3519}$, $\frac{1}{10557} a^{11} + \frac{8}{621} a^{9} + \frac{191}{10557} a^{8} + \frac{2495}{10557} a^{7} + \frac{2701}{10557} a^{6} + \frac{3260}{10557} a^{5} - \frac{1511}{10557} a^{4} + \frac{6}{391} a^{3} - \frac{1420}{3519} a^{2} + \frac{4865}{10557} a - \frac{676}{3519}$, $\frac{1}{10557} a^{12} + \frac{89}{10557} a^{9} + \frac{149}{10557} a^{8} + \frac{967}{10557} a^{7} + \frac{268}{10557} a^{6} + \frac{3878}{10557} a^{5} - \frac{76}{153} a^{4} + \frac{4223}{10557} a^{3} + \frac{2978}{10557} a^{2} + \frac{4177}{10557} a - \frac{32}{69}$, $\frac{1}{10557} a^{13} - \frac{188}{10557} a^{9} - \frac{4}{621} a^{8} + \frac{5257}{10557} a^{7} + \frac{157}{3519} a^{6} + \frac{3265}{10557} a^{5} - \frac{1441}{3519} a^{4} + \frac{500}{3519} a^{3} + \frac{353}{3519} a^{2} + \frac{5090}{10557} a + \frac{5053}{10557}$, $\frac{1}{8402310144618394143573} a^{14} - \frac{1}{1200330020659770591939} a^{13} - \frac{336889861960790345}{8402310144618394143573} a^{12} - \frac{366359079050374106}{8402310144618394143573} a^{11} - \frac{65122378452941578}{8402310144618394143573} a^{10} - \frac{3525957844836787511}{933590016068710460397} a^{9} - \frac{72649301398341648863}{8402310144618394143573} a^{8} + \frac{549089073284901848948}{1200330020659770591939} a^{7} - \frac{14108722333761005905}{35156109391708762107} a^{6} + \frac{3975600319593354676550}{8402310144618394143573} a^{5} - \frac{435511255835767050071}{8402310144618394143573} a^{4} - \frac{69971318338708085615}{2800770048206131381191} a^{3} + \frac{3763152316058381984092}{8402310144618394143573} a^{2} - \frac{3628355213562617806124}{8402310144618394143573} a - \frac{2430997033089096207595}{8402310144618394143573}$, $\frac{1}{435162372640212472029354671973} a^{15} + \frac{25895393}{435162372640212472029354671973} a^{14} + \frac{169798780060778566563814}{6306701052756702493179053217} a^{13} + \frac{2596002862793510821566266}{62166053234316067432764953139} a^{12} + \frac{19880439880166239529908072}{435162372640212472029354671973} a^{11} + \frac{1555182968488010754326165}{145054124213404157343118223991} a^{10} + \frac{528972597025533754742697943}{62166053234316067432764953139} a^{9} - \frac{4275533604348861700892937722}{435162372640212472029354671973} a^{8} + \frac{81894968645484448264838285822}{435162372640212472029354671973} a^{7} + \frac{16941067300246463008695858125}{62166053234316067432764953139} a^{6} + \frac{1067097916122196265236165546}{4679165297206585720745749161} a^{5} - \frac{11633570833233826682743059878}{25597786625894851295844392469} a^{4} - \frac{24351259623236780573256381487}{145054124213404157343118223991} a^{3} + \frac{47743272679534031957014242685}{145054124213404157343118223991} a^{2} - \frac{13371277683789567979558980862}{435162372640212472029354671973} a + \frac{169801752329822676643357382702}{435162372640212472029354671973}$

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

Class group and class number

$C_{6}\times C_{360528}$, which has order $2163168$ (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

\(\Q(\sqrt{-53}) \), \(\Q(\sqrt{-530}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-106}) \), \(\Q(\sqrt{-265}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}, \sqrt{-53})\), \(\Q(\sqrt{2}, \sqrt{-53})\), \(\Q(\sqrt{5}, \sqrt{-53})\), \(\Q(\sqrt{2}, \sqrt{-265})\), \(\Q(\sqrt{5}, \sqrt{-106})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{10}, \sqrt{-106})\), 4.0.5752832.2, \(\Q(\zeta_{16})^+\), 4.4.51200.1, 4.0.143820800.2, 8.0.323194101760000.29, 8.0.132380304080896.14, 8.0.82737690050560000.4, 8.0.82737690050560000.3, 8.0.82737690050560000.8, 8.0.20684422512640000.81, 8.8.2621440000.1

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
$53$53.8.4.1$x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
53.8.4.1$x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$