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Properties

Label 16.0.65284017132...0000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 53^{8}$
Root discriminant $97.37$
Ramified primes $2, 5, 53$
Class number $171360$ (GRH)
Class group $[6, 28560]$ (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![4571966921, 775588320, 1780212660, 194588720, 298724599, 18829560, 29864470, 925040, 2045996, 22440, 99940, -160, 3434, -20, 80, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 80*x^14 - 20*x^13 + 3434*x^12 - 160*x^11 + 99940*x^10 + 22440*x^9 + 2045996*x^8 + 925040*x^7 + 29864470*x^6 + 18829560*x^5 + 298724599*x^4 + 194588720*x^3 + 1780212660*x^2 + 775588320*x + 4571966921)
 
gp: K = bnfinit(x^16 + 80*x^14 - 20*x^13 + 3434*x^12 - 160*x^11 + 99940*x^10 + 22440*x^9 + 2045996*x^8 + 925040*x^7 + 29864470*x^6 + 18829560*x^5 + 298724599*x^4 + 194588720*x^3 + 1780212660*x^2 + 775588320*x + 4571966921, 1)
 

Normalized defining polynomial

\( x^{16} + 80 x^{14} - 20 x^{13} + 3434 x^{12} - 160 x^{11} + 99940 x^{10} + 22440 x^{9} + 2045996 x^{8} + 925040 x^{7} + 29864470 x^{6} + 18829560 x^{5} + 298724599 x^{4} + 194588720 x^{3} + 1780212660 x^{2} + 775588320 x + 4571966921 \)

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:  \(65284017132783271936000000000000=2^{32}\cdot 5^{12}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.37$
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:  \(2120=2^{3}\cdot 5\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{2120}(1,·)$, $\chi_{2120}(2119,·)$, $\chi_{2120}(1803,·)$, $\chi_{2120}(1167,·)$, $\chi_{2120}(849,·)$, $\chi_{2120}(211,·)$, $\chi_{2120}(2013,·)$, $\chi_{2120}(1377,·)$, $\chi_{2120}(1059,·)$, $\chi_{2120}(1061,·)$, $\chi_{2120}(743,·)$, $\chi_{2120}(107,·)$, $\chi_{2120}(1909,·)$, $\chi_{2120}(1271,·)$, $\chi_{2120}(953,·)$, $\chi_{2120}(317,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{123} a^{14} + \frac{10}{123} a^{13} - \frac{4}{41} a^{12} - \frac{10}{123} a^{11} - \frac{61}{123} a^{10} - \frac{13}{41} a^{9} - \frac{4}{41} a^{8} + \frac{14}{41} a^{7} + \frac{32}{123} a^{6} - \frac{38}{123} a^{5} - \frac{58}{123} a^{4} - \frac{26}{123} a^{3} + \frac{15}{41} a^{2} + \frac{44}{123} a + \frac{40}{123}$, $\frac{1}{345436920301932405604093670765781518491401242823} a^{15} + \frac{757517972358247986568715352340376569436373793}{345436920301932405604093670765781518491401242823} a^{14} + \frac{5023738198106977502812122023332376252704171063}{115145640100644135201364556921927172830467080941} a^{13} - \frac{31451725269195560081187854115133407862568078161}{345436920301932405604093670765781518491401242823} a^{12} - \frac{293892364658254492755204184299852169138371319}{1433348217020466413294994484505317504113698103} a^{11} + \frac{24648899913755928767101959641385640436890623938}{115145640100644135201364556921927172830467080941} a^{10} - \frac{54798254736384379663262103061470206007073469872}{115145640100644135201364556921927172830467080941} a^{9} + \frac{42566226367929967856236128612019590000506819893}{115145640100644135201364556921927172830467080941} a^{8} + \frac{155354876655548122473091729707186787448644421084}{345436920301932405604093670765781518491401242823} a^{7} + \frac{165798041974041772279222029732652194463816231642}{345436920301932405604093670765781518491401242823} a^{6} + \frac{148165867772666973703466196714306885988652863964}{345436920301932405604093670765781518491401242823} a^{5} - \frac{114094631141937799842204708390099505469570825939}{345436920301932405604093670765781518491401242823} a^{4} - \frac{11105240061320271518823203951095215658720857865}{115145640100644135201364556921927172830467080941} a^{3} - \frac{13488537498294607866346222755992637348523729429}{345436920301932405604093670765781518491401242823} a^{2} - \frac{141186091270551732178430723547159963802903345715}{345436920301932405604093670765781518491401242823} a + \frac{34848139603844343366823012918905889981626744}{1457539748109419432928665277492749023170469379}$

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

Class group and class number

$C_{6}\times C_{28560}$, which has order $171360$ (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:  \( 7114.135357253273 \) (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

\(\Q(\sqrt{-53}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-106}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-265}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-530}) \), \(\Q(\sqrt{2}, \sqrt{-53})\), \(\Q(\sqrt{5}, \sqrt{-53})\), \(\Q(\sqrt{10}, \sqrt{-53})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-265})\), \(\Q(\sqrt{5}, \sqrt{-106})\), \(\Q(\sqrt{10}, \sqrt{-106})\), 4.0.351125.1, \(\Q(\zeta_{20})^+\), 4.0.22472000.2, 4.4.8000.1, 8.0.323194101760000.29, 8.0.31561924000000.13, 8.0.8079852544000000.33, 8.0.504990784000000.34, \(\Q(\zeta_{40})^+\), 8.0.504990784000000.33, 8.0.8079852544000000.23

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$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}$
$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}$