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Properties

Label 16.0.31451492021...000.28
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 17^{8}$
Root discriminant $191.03$
Ramified primes $2, 3, 5, 17$
Class number $12847104$ (GRH)
Class group $[2, 2, 4, 24, 33456]$ (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![571570506, 777914280, 756378756, 481832280, 230936022, 77418000, 15990588, 374400, -899765, -245400, 7772, 15960, 2382, -240, -76, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 76*x^14 - 240*x^13 + 2382*x^12 + 15960*x^11 + 7772*x^10 - 245400*x^9 - 899765*x^8 + 374400*x^7 + 15990588*x^6 + 77418000*x^5 + 230936022*x^4 + 481832280*x^3 + 756378756*x^2 + 777914280*x + 571570506)
 
gp: K = bnfinit(x^16 - 76*x^14 - 240*x^13 + 2382*x^12 + 15960*x^11 + 7772*x^10 - 245400*x^9 - 899765*x^8 + 374400*x^7 + 15990588*x^6 + 77418000*x^5 + 230936022*x^4 + 481832280*x^3 + 756378756*x^2 + 777914280*x + 571570506, 1)
 

Normalized defining polynomial

\( x^{16} - 76 x^{14} - 240 x^{13} + 2382 x^{12} + 15960 x^{11} + 7772 x^{10} - 245400 x^{9} - 899765 x^{8} + 374400 x^{7} + 15990588 x^{6} + 77418000 x^{5} + 230936022 x^{4} + 481832280 x^{3} + 756378756 x^{2} + 777914280 x + 571570506 \)

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:  \(3145149202160209033691136000000000000=2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $191.03$
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:  $2, 3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4080=2^{4}\cdot 3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{4080}(1,·)$, $\chi_{4080}(3911,·)$, $\chi_{4080}(203,·)$, $\chi_{4080}(271,·)$, $\chi_{4080}(1427,·)$, $\chi_{4080}(409,·)$, $\chi_{4080}(3161,·)$, $\chi_{4080}(1123,·)$, $\chi_{4080}(679,·)$, $\chi_{4080}(2347,·)$, $\chi_{4080}(2413,·)$, $\chi_{4080}(239,·)$, $\chi_{4080}(3569,·)$, $\chi_{4080}(3637,·)$, $\chi_{4080}(3197,·)$, $\chi_{4080}(1973,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{15} a^{8} + \frac{2}{15} a^{6} - \frac{1}{15} a^{4} - \frac{2}{15} a^{2} + \frac{2}{5}$, $\frac{1}{45} a^{9} - \frac{1}{15} a^{7} - \frac{2}{15} a^{5} - \frac{7}{45} a^{3} + \frac{2}{15} a$, $\frac{1}{45} a^{10} - \frac{2}{9} a^{4} + \frac{2}{5}$, $\frac{1}{45} a^{11} - \frac{2}{9} a^{5} + \frac{2}{5} a$, $\frac{1}{225} a^{12} - \frac{2}{225} a^{10} + \frac{2}{15} a^{7} - \frac{2}{45} a^{6} - \frac{7}{15} a^{5} + \frac{4}{45} a^{4} - \frac{1}{15} a^{3} + \frac{12}{25} a^{2} - \frac{2}{5} a + \frac{1}{25}$, $\frac{1}{47925} a^{13} - \frac{32}{15975} a^{12} - \frac{172}{47925} a^{11} + \frac{49}{15975} a^{10} - \frac{28}{3195} a^{9} - \frac{64}{3195} a^{8} - \frac{701}{9585} a^{7} - \frac{14}{213} a^{6} + \frac{1859}{9585} a^{5} - \frac{132}{355} a^{4} - \frac{5804}{15975} a^{3} + \frac{1058}{5325} a^{2} - \frac{3}{1775} a - \frac{557}{1775}$, $\frac{1}{411796473075} a^{14} - \frac{1431728}{137265491025} a^{13} - \frac{389872537}{411796473075} a^{12} - \frac{55057227}{5083907075} a^{11} - \frac{98826181}{9151032735} a^{10} - \frac{64013059}{27453098205} a^{9} - \frac{1375478954}{82359294615} a^{8} + \frac{704284274}{9151032735} a^{7} + \frac{4740635807}{82359294615} a^{6} + \frac{9147020933}{27453098205} a^{5} - \frac{29791939414}{137265491025} a^{4} - \frac{4484492648}{45755163675} a^{3} - \frac{2997003134}{45755163675} a^{2} - \frac{1039400956}{5083907075} a - \frac{5705096}{15642791}$, $\frac{1}{185106395228307125977718739128925} a^{15} - \frac{163091910582718727}{4113475449517936132838194202865} a^{14} - \frac{127344360041997592263224344}{185106395228307125977718739128925} a^{13} - \frac{74970660828344435281876127606}{61702131742769041992572913042975} a^{12} - \frac{73833055264918490844602363653}{6855792415863226888063657004775} a^{11} + \frac{70649247923198082110888277832}{61702131742769041992572913042975} a^{10} + \frac{320895964908737314238865582424}{37021279045661425195543747825785} a^{9} + \frac{21145153151421339038943876310}{2468085269710761679702916521719} a^{8} + \frac{3554843278486447017054470730572}{37021279045661425195543747825785} a^{7} + \frac{380243214556499296946271374114}{4113475449517936132838194202865} a^{6} - \frac{771103524087578063554555458199}{61702131742769041992572913042975} a^{5} - \frac{108064541414096540996518065713}{1371158483172645377612731400955} a^{4} + \frac{564992737184977107247409525287}{20567377247589680664190971014325} a^{3} - \frac{751289510084599223338369904749}{2285264138621075629354552334925} a^{2} - \frac{529251162443502231109775006233}{2285264138621075629354552334925} a + \frac{4125854947039858996117754277}{58596516374899375111655188075}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{24}\times C_{33456}$, which has order $12847104$ (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:  \( 2699811.4861512356 \) (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

\(\Q(\sqrt{-170}) \), \(\Q(\sqrt{-255}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{-102}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{6}, \sqrt{-170})\), \(\Q(\sqrt{10}, \sqrt{-17})\), \(\Q(\sqrt{15}, \sqrt{-102})\), \(\Q(\sqrt{10}, \sqrt{-102})\), \(\Q(\sqrt{15}, \sqrt{-17})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{6}, \sqrt{-17})\), 4.0.665856000.4, 4.4.2304000.2, 4.4.256000.1, 4.0.73984000.2, 8.0.277102632960000.73, 8.0.1773456850944000000.14, 8.0.21894529024000000.37, 8.0.443364212736000000.15, 8.0.443364212736000000.12, 8.0.1773456850944000000.19, 8.8.21233664000000.3

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.24.13$x^{8} + 28 x^{4} + 36$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.13$x^{8} + 28 x^{4} + 36$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$