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Properties

Label 16.0.43973667993...2176.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 47^{8}$
Root discriminant $94.99$
Ramified primes $2, 3, 47$
Class number $409600$ (GRH)
Class group $[2, 2, 2, 4, 8, 40, 40]$ (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![1365774814, -595814888, 718599116, -261158920, 170888666, -52182408, 23986068, -6139592, 2169447, -459208, 129180, -21864, 4918, -616, 108, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 108*x^14 - 616*x^13 + 4918*x^12 - 21864*x^11 + 129180*x^10 - 459208*x^9 + 2169447*x^8 - 6139592*x^7 + 23986068*x^6 - 52182408*x^5 + 170888666*x^4 - 261158920*x^3 + 718599116*x^2 - 595814888*x + 1365774814)
 
gp: K = bnfinit(x^16 - 8*x^15 + 108*x^14 - 616*x^13 + 4918*x^12 - 21864*x^11 + 129180*x^10 - 459208*x^9 + 2169447*x^8 - 6139592*x^7 + 23986068*x^6 - 52182408*x^5 + 170888666*x^4 - 261158920*x^3 + 718599116*x^2 - 595814888*x + 1365774814, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 108 x^{14} - 616 x^{13} + 4918 x^{12} - 21864 x^{11} + 129180 x^{10} - 459208 x^{9} + 2169447 x^{8} - 6139592 x^{7} + 23986068 x^{6} - 52182408 x^{5} + 170888666 x^{4} - 261158920 x^{3} + 718599116 x^{2} - 595814888 x + 1365774814 \)

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:  \(43973667993577319434061585842176=2^{48}\cdot 3^{8}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $94.99$
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:  $2, 3, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2256=2^{4}\cdot 3\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{2256}(1,·)$, $\chi_{2256}(1223,·)$, $\chi_{2256}(1033,·)$, $\chi_{2256}(2255,·)$, $\chi_{2256}(659,·)$, $\chi_{2256}(469,·)$, $\chi_{2256}(1691,·)$, $\chi_{2256}(1693,·)$, $\chi_{2256}(95,·)$, $\chi_{2256}(1127,·)$, $\chi_{2256}(1129,·)$, $\chi_{2256}(2161,·)$, $\chi_{2256}(563,·)$, $\chi_{2256}(565,·)$, $\chi_{2256}(1787,·)$, $\chi_{2256}(1597,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{298903403047874259977} a^{14} - \frac{7}{298903403047874259977} a^{13} - \frac{9276848494733540721}{298903403047874259977} a^{12} + \frac{55661090968401244417}{298903403047874259977} a^{11} - \frac{89382536612221574297}{298903403047874259977} a^{10} - \frac{63313984149236869171}{298903403047874259977} a^{9} + \frac{41213040278844815181}{298903403047874259977} a^{8} - \frac{69406464711167136756}{298903403047874259977} a^{7} - \frac{7362051250028451796}{298903403047874259977} a^{6} + \frac{134076553779647444571}{298903403047874259977} a^{5} + \frac{17691064046336745247}{298903403047874259977} a^{4} + \frac{142763942577545401789}{298903403047874259977} a^{3} + \frac{38407925458714760760}{298903403047874259977} a^{2} + \frac{107831671155771420759}{298903403047874259977} a + \frac{92573041183470389649}{298903403047874259977}$, $\frac{1}{2677361772956066199281042537} a^{15} + \frac{4478633}{2677361772956066199281042537} a^{14} + \frac{1186736769650932775370311853}{2677361772956066199281042537} a^{13} + \frac{2853078832538741088153121}{27601667762433672157536521} a^{12} + \frac{945239132075717209662926418}{2677361772956066199281042537} a^{11} - \frac{1035909430430035443389277361}{2677361772956066199281042537} a^{10} + \frac{13866734961689545334643781}{116407033606785486925262719} a^{9} - \frac{539868496397897150274478230}{2677361772956066199281042537} a^{8} + \frac{18895755214850477555793722}{116407033606785486925262719} a^{7} + \frac{665797059926370602413538306}{2677361772956066199281042537} a^{6} - \frac{2619464747911008805664062}{27601667762433672157536521} a^{5} - \frac{154514595986652163263227809}{2677361772956066199281042537} a^{4} - \frac{32050029305735420299426483}{2677361772956066199281042537} a^{3} + \frac{369731610928972914397093303}{2677361772956066199281042537} a^{2} - \frac{1164094941786842128945346703}{2677361772956066199281042537} a + \frac{1244649798458420640494266843}{2677361772956066199281042537}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{8}\times C_{40}\times C_{40}$, which has order $409600$ (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:  \( 11964.310642723332 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-47}) \), \(\Q(\sqrt{-141}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-94}) \), \(\Q(\sqrt{-282}) \), \(\Q(\sqrt{3}, \sqrt{-47})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-94})\), \(\Q(\sqrt{2}, \sqrt{-47})\), \(\Q(\sqrt{6}, \sqrt{-47})\), \(\Q(\sqrt{2}, \sqrt{-141})\), \(\Q(\sqrt{6}, \sqrt{-94})\), \(\Q(\zeta_{16})^+\), 4.4.18432.1, 4.0.4524032.5, 4.0.40716288.5, 8.0.25903376695296.46, \(\Q(\zeta_{48})^+\), 8.0.6631264433995776.65, 8.0.20466865537024.21, 8.0.1657816108498944.53, 8.0.6631264433995776.69, 8.0.6631264433995776.49

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 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}$
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$