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Properties

Label 16.0.22538752331...2544.8
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 73^{14}$
Root discriminant $591.64$
Ramified primes $2, 3, 73$
Class number $1184370688$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 4, 2313224]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18734319395856, 0, 7803495656064, 0, 912726990288, 0, 46480731696, 0, 1211607937, 0, 17392688, 0, 139284, 0, 584, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 584*x^14 + 139284*x^12 + 17392688*x^10 + 1211607937*x^8 + 46480731696*x^6 + 912726990288*x^4 + 7803495656064*x^2 + 18734319395856)
 
gp: K = bnfinit(x^16 + 584*x^14 + 139284*x^12 + 17392688*x^10 + 1211607937*x^8 + 46480731696*x^6 + 912726990288*x^4 + 7803495656064*x^2 + 18734319395856, 1)
 

Normalized defining polynomial

\( x^{16} + 584 x^{14} + 139284 x^{12} + 17392688 x^{10} + 1211607937 x^{8} + 46480731696 x^{6} + 912726990288 x^{4} + 7803495656064 x^{2} + 18734319395856 \)

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:  \(225387523315734662028879718967317456742252544=2^{48}\cdot 3^{8}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $591.64$
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:  $2, 3, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3504=2^{4}\cdot 3\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{3504}(1,·)$, $\chi_{3504}(265,·)$, $\chi_{3504}(971,·)$, $\chi_{3504}(145,·)$, $\chi_{3504}(83,·)$, $\chi_{3504}(3421,·)$, $\chi_{3504}(3359,·)$, $\chi_{3504}(2533,·)$, $\chi_{3504}(3239,·)$, $\chi_{3504}(3503,·)$, $\chi_{3504}(1523,·)$, $\chi_{3504}(2869,·)$, $\chi_{3504}(119,·)$, $\chi_{3504}(3385,·)$, $\chi_{3504}(635,·)$, $\chi_{3504}(1981,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{657} a^{8} + \frac{1}{3} a^{4} - \frac{4}{9} a^{2}$, $\frac{1}{657} a^{9} + \frac{2}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{1314} a^{10} + \frac{1}{9} a^{4} - \frac{1}{6} a^{2}$, $\frac{1}{3942} a^{11} + \frac{1}{1971} a^{9} + \frac{4}{27} a^{5} + \frac{7}{54} a^{3} + \frac{1}{3} a$, $\frac{1}{70956} a^{12} + \frac{13}{35478} a^{10} + \frac{1}{1971} a^{8} + \frac{29}{243} a^{6} - \frac{431}{972} a^{4} - \frac{23}{54} a^{2}$, $\frac{1}{25969896} a^{13} + \frac{749}{6492474} a^{11} - \frac{88}{360693} a^{9} + \frac{5644}{44469} a^{7} + \frac{42121}{355752} a^{5} + \frac{2591}{9882} a^{3} + \frac{19}{122} a$, $\frac{1}{132950987181659799515575224} a^{14} + \frac{2555446453391324497}{910623199874382188462844} a^{12} + \frac{192198454318489653737}{1846541488634163882160767} a^{10} + \frac{7571047365364697295103}{16618873397707474939446903} a^{8} - \frac{252960459343507227225671}{1821246399748764376925688} a^{6} - \frac{38216737915913032287119}{101180355541598020940316} a^{4} + \frac{20108862900987962837}{624570095935790252718} a^{2} + \frac{147125676757907966}{1706475671955711073}$, $\frac{1}{2393117769269876391280354032} a^{15} + \frac{8803249786975646027}{598279442317469097820088508} a^{13} - \frac{294766772971074481}{3732062294567140116651} a^{11} - \frac{26831209585711041568678}{149569860579367274455022127} a^{9} - \frac{2153209856801133203986967}{32782435195477758784662384} a^{7} - \frac{15339617436442328238839}{910623199874382188462844} a^{5} - \frac{19644854590672559580581}{101180355541598020940316} a^{3} + \frac{42029797924141836569}{936855143903685379077} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{2313224}$, which has order $1184370688$ (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:  \( 148350354.74649996 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

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 $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{219}) \), \(\Q(\sqrt{73}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{3}, \sqrt{73})\), 4.4.24897088.1, 4.4.224073792.2, 8.8.803345028180148224.2, 8.0.46336147798242623488.23, 8.0.3753227971657652502528.1

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$73$73.8.7.1$x^{8} - 73$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.1$x^{8} - 73$$8$$1$$7$$C_8$$[\ ]_{8}$