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Properties

Label 16.0.36778727586...000.15
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{8}\cdot 5^{8}\cdot 13^{8}$
Root discriminant $93.94$
Ramified primes $2, 3, 5, 13$
Class number $119808$ (GRH)
Class group $[4, 24, 1248]$ (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![606644476, -278944192, 338600600, -127019952, 83442994, -25822592, 11902356, -3076240, 1097171, -239384, 69340, -12768, 3038, -448, 84, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 84*x^14 - 448*x^13 + 3038*x^12 - 12768*x^11 + 69340*x^10 - 239384*x^9 + 1097171*x^8 - 3076240*x^7 + 11902356*x^6 - 25822592*x^5 + 83442994*x^4 - 127019952*x^3 + 338600600*x^2 - 278944192*x + 606644476)
 
gp: K = bnfinit(x^16 - 8*x^15 + 84*x^14 - 448*x^13 + 3038*x^12 - 12768*x^11 + 69340*x^10 - 239384*x^9 + 1097171*x^8 - 3076240*x^7 + 11902356*x^6 - 25822592*x^5 + 83442994*x^4 - 127019952*x^3 + 338600600*x^2 - 278944192*x + 606644476, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 84 x^{14} - 448 x^{13} + 3038 x^{12} - 12768 x^{11} + 69340 x^{10} - 239384 x^{9} + 1097171 x^{8} - 3076240 x^{7} + 11902356 x^{6} - 25822592 x^{5} + 83442994 x^{4} - 127019952 x^{3} + 338600600 x^{2} - 278944192 x + 606644476 \)

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:  \(36778727586648064367001600000000=2^{44}\cdot 3^{8}\cdot 5^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.94$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3120=2^{4}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{3120}(1,·)$, $\chi_{3120}(389,·)$, $\chi_{3120}(1249,·)$, $\chi_{3120}(1481,·)$, $\chi_{3120}(781,·)$, $\chi_{3120}(1169,·)$, $\chi_{3120}(2261,·)$, $\chi_{3120}(1561,·)$, $\chi_{3120}(1949,·)$, $\chi_{3120}(3041,·)$, $\chi_{3120}(2341,·)$, $\chi_{3120}(2729,·)$, $\chi_{3120}(2029,·)$, $\chi_{3120}(2809,·)$, $\chi_{3120}(701,·)$, $\chi_{3120}(469,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{124874325034557428258} a^{14} - \frac{7}{124874325034557428258} a^{13} - \frac{13221215906054423196}{62437162517278714129} a^{12} - \frac{14328448339591531972}{62437162517278714129} a^{11} + \frac{3648367798085017238}{62437162517278714129} a^{10} + \frac{333672729037061485}{5429318479763366446} a^{9} + \frac{1299449952016074430}{62437162517278714129} a^{8} + \frac{1477337704707750653}{62437162517278714129} a^{7} - \frac{34003001019190075701}{124874325034557428258} a^{6} + \frac{22886082180876958027}{62437162517278714129} a^{5} + \frac{25738220393679831722}{62437162517278714129} a^{4} + \frac{12934198482901418068}{62437162517278714129} a^{3} - \frac{14278264803108444806}{62437162517278714129} a^{2} - \frac{12991463337843819388}{62437162517278714129} a - \frac{105219211843607409}{790343829332641951}$, $\frac{1}{743222137642002553766262338} a^{15} + \frac{2975873}{743222137642002553766262338} a^{14} + \frac{151829982579897768985664973}{743222137642002553766262338} a^{13} - \frac{136556731919066686189845007}{743222137642002553766262338} a^{12} + \frac{42934797776182518353266353}{743222137642002553766262338} a^{11} - \frac{139827830014326517770541311}{743222137642002553766262338} a^{10} + \frac{330717144366702197044058}{5090562586589058587440153} a^{9} + \frac{149981545143239348308106485}{743222137642002553766262338} a^{8} + \frac{143516257105941155173813236}{371611068821001276883131169} a^{7} + \frac{3664938900376829916018309}{16157002992217446821005703} a^{6} + \frac{16389548681875002617619451}{743222137642002553766262338} a^{5} - \frac{18441487079557441134610956}{371611068821001276883131169} a^{4} + \frac{58726430237719382862249805}{371611068821001276883131169} a^{3} + \frac{65730919142861222167270968}{371611068821001276883131169} a^{2} + \frac{52781364571702657170994328}{371611068821001276883131169} a + \frac{1522937485907458091776592}{4703937580012674390925711}$

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

Class group and class number

$C_{4}\times C_{24}\times C_{1248}$, which has order $119808$ (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

\(\Q(\sqrt{-195}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-78}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-390}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{10}, \sqrt{-78})\), \(\Q(\sqrt{2}, \sqrt{-195})\), \(\Q(\sqrt{5}, \sqrt{-39})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{10}, \sqrt{-39})\), \(\Q(\sqrt{2}, \sqrt{-39})\), \(\Q(\sqrt{5}, \sqrt{-78})\), 4.0.77875200.2, \(\Q(\zeta_{16})^+\), 4.0.3115008.1, 4.4.51200.1, 8.0.5922408960000.17, 8.0.6064546775040000.428, 8.0.6064546775040000.225, 8.0.6064546775040000.517, 8.8.2621440000.1, 8.0.6064546775040000.285, 8.0.9703274840064.5

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$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$5.8.4.1$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}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$