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Properties

Label 16.0.28017318032...8976.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 79^{8}$
Root discriminant $123.16$
Ramified primes $2, 3, 79$
Class number $2611200$ (GRH)
Class group $[2, 4, 80, 4080]$ (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![53156046238, -16229095208, 18507674124, -4749147848, 2865167258, -618838472, 257685716, -46483976, 14717223, -2173192, 545756, -63272, 12790, -1064, 172, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 172*x^14 - 1064*x^13 + 12790*x^12 - 63272*x^11 + 545756*x^10 - 2173192*x^9 + 14717223*x^8 - 46483976*x^7 + 257685716*x^6 - 618838472*x^5 + 2865167258*x^4 - 4749147848*x^3 + 18507674124*x^2 - 16229095208*x + 53156046238)
 
gp: K = bnfinit(x^16 - 8*x^15 + 172*x^14 - 1064*x^13 + 12790*x^12 - 63272*x^11 + 545756*x^10 - 2173192*x^9 + 14717223*x^8 - 46483976*x^7 + 257685716*x^6 - 618838472*x^5 + 2865167258*x^4 - 4749147848*x^3 + 18507674124*x^2 - 16229095208*x + 53156046238, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 172 x^{14} - 1064 x^{13} + 12790 x^{12} - 63272 x^{11} + 545756 x^{10} - 2173192 x^{9} + 14717223 x^{8} - 46483976 x^{7} + 257685716 x^{6} - 618838472 x^{5} + 2865167258 x^{4} - 4749147848 x^{3} + 18507674124 x^{2} - 16229095208 x + 53156046238 \)

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:  \(2801731803266966736034164664958976=2^{48}\cdot 3^{8}\cdot 79^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $123.16$
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:  $2, 3, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3792=2^{4}\cdot 3\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{3792}(1,·)$, $\chi_{3792}(2053,·)$, $\chi_{3792}(1739,·)$, $\chi_{3792}(3791,·)$, $\chi_{3792}(1105,·)$, $\chi_{3792}(791,·)$, $\chi_{3792}(2843,·)$, $\chi_{3792}(157,·)$, $\chi_{3792}(1895,·)$, $\chi_{3792}(1897,·)$, $\chi_{3792}(2845,·)$, $\chi_{3792}(3635,·)$, $\chi_{3792}(947,·)$, $\chi_{3792}(949,·)$, $\chi_{3792}(3001,·)$, $\chi_{3792}(2687,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{105453688754156979387529} a^{14} - \frac{1}{15064812679165282769647} a^{13} + \frac{42465322670569725299880}{105453688754156979387529} a^{12} - \frac{43884558515104393024131}{105453688754156979387529} a^{11} + \frac{6734475118775541366657}{105453688754156979387529} a^{10} - \frac{18060781303996361866524}{105453688754156979387529} a^{9} - \frac{32179725803635929822724}{105453688754156979387529} a^{8} - \frac{4962739302758924857151}{15064812679165282769647} a^{7} + \frac{35041414520391094879823}{105453688754156979387529} a^{6} + \frac{43704452030818851991366}{105453688754156979387529} a^{5} + \frac{14818825739392650695126}{105453688754156979387529} a^{4} + \frac{3261344119427219016314}{105453688754156979387529} a^{3} + \frac{48691056872438970284041}{105453688754156979387529} a^{2} + \frac{39601038424392084567764}{105453688754156979387529} a - \frac{26222377316423162665104}{105453688754156979387529}$, $\frac{1}{5990360167346828462453196749929} a^{15} + \frac{28402793}{5990360167346828462453196749929} a^{14} + \frac{1097195372542790308217597873947}{5990360167346828462453196749929} a^{13} - \frac{418636535696343377593826545580}{855765738192404066064742392847} a^{12} + \frac{894871908561391148788070391607}{5990360167346828462453196749929} a^{11} + \frac{969545769234563423410653673470}{5990360167346828462453196749929} a^{10} + \frac{146481771821602609121413980}{913303882809396015010397431} a^{9} + \frac{2228428873046604716916926608650}{5990360167346828462453196749929} a^{8} - \frac{1084484184469958865798378073449}{5990360167346828462453196749929} a^{7} + \frac{179644048550047281795429083672}{855765738192404066064742392847} a^{6} - \frac{1149330975260139766715438191860}{5990360167346828462453196749929} a^{5} - \frac{165581695784481980034555301103}{5990360167346828462453196749929} a^{4} - \frac{2878926523950870631264871684314}{5990360167346828462453196749929} a^{3} - \frac{144127978803646070011133133199}{5990360167346828462453196749929} a^{2} + \frac{719272740442883592566537383598}{5990360167346828462453196749929} a + \frac{2363046213662844409113596145410}{5990360167346828462453196749929}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{80}\times C_{4080}$, which has order $2611200$ (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-474}) \), \(\Q(\sqrt{-158}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-237}) \), \(\Q(\sqrt{-79}) \), \(\Q(\sqrt{3}, \sqrt{-158})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-79})\), \(\Q(\sqrt{2}, \sqrt{-237})\), \(\Q(\sqrt{6}, \sqrt{-79})\), \(\Q(\sqrt{2}, \sqrt{-79})\), \(\Q(\sqrt{6}, \sqrt{-158})\), 4.0.12781568.9, 4.0.115034112.5, 4.4.18432.1, \(\Q(\zeta_{16})^+\), 8.0.206763233181696.27, 8.0.52931387694514176.31, \(\Q(\zeta_{48})^+\), 8.0.52931387694514176.17, 8.0.52931387694514176.35, 8.0.163368480538624.33, 8.0.13232846923628544.65

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.24.9$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}$
$79$79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$