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Properties

Label 16.0.26268600651...000.76
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 17^{12}$
Root discriminant $387.90$
Ramified primes $2, 3, 5, 17$
Class number $112919040$ (GRH)
Class group $[2, 2, 2, 4, 156, 22620]$ (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![1082432160000, 0, 721621440000, 0, 157766256000, 0, 14090484000, 0, 615403825, 0, 13814200, 0, 151640, 0, 680, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 680*x^14 + 151640*x^12 + 13814200*x^10 + 615403825*x^8 + 14090484000*x^6 + 157766256000*x^4 + 721621440000*x^2 + 1082432160000)
 
gp: K = bnfinit(x^16 + 680*x^14 + 151640*x^12 + 13814200*x^10 + 615403825*x^8 + 14090484000*x^6 + 157766256000*x^4 + 721621440000*x^2 + 1082432160000, 1)
 

Normalized defining polynomial

\( x^{16} + 680 x^{14} + 151640 x^{12} + 13814200 x^{10} + 615403825 x^{8} + 14090484000 x^{6} + 157766256000 x^{4} + 721621440000 x^{2} + 1082432160000 \)

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:  \(262686006513622818702917369856000000000000=2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $387.90$
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:  $2, 3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4080=2^{4}\cdot 3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{4080}(1,·)$, $\chi_{4080}(1067,·)$, $\chi_{4080}(2053,·)$, $\chi_{4080}(3841,·)$, $\chi_{4080}(3277,·)$, $\chi_{4080}(3911,·)$, $\chi_{4080}(3923,·)$, $\chi_{4080}(3671,·)$, $\chi_{4080}(409,·)$, $\chi_{4080}(4079,·)$, $\chi_{4080}(157,·)$, $\chi_{4080}(3013,·)$, $\chi_{4080}(803,·)$, $\chi_{4080}(169,·)$, $\chi_{4080}(2027,·)$, $\chi_{4080}(239,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{85} a^{4}$, $\frac{1}{85} a^{5}$, $\frac{1}{85} a^{6}$, $\frac{1}{85} a^{7}$, $\frac{1}{7225} a^{8}$, $\frac{1}{14450} a^{9} - \frac{1}{2} a$, $\frac{1}{433500} a^{10} + \frac{1}{21675} a^{8} - \frac{7}{1275} a^{6} - \frac{1}{255} a^{4} - \frac{11}{60} a^{2}$, $\frac{1}{867000} a^{11} + \frac{1}{43350} a^{9} + \frac{4}{1275} a^{7} + \frac{1}{255} a^{5} + \frac{49}{120} a^{3} - \frac{1}{2} a$, $\frac{1}{156616614000} a^{12} + \frac{1987}{2303185500} a^{10} + \frac{947}{46063710} a^{8} + \frac{19507}{13548150} a^{6} + \frac{105673}{21677040} a^{4} + \frac{33391}{106260} a^{2} + \frac{12}{1771}$, $\frac{1}{939699684000} a^{13} + \frac{1987}{13819113000} a^{11} - \frac{27143}{1381911300} a^{9} - \frac{299273}{81288900} a^{7} - \frac{149351}{130062240} a^{5} + \frac{139651}{637560} a^{3} - \frac{1759}{10626} a$, $\frac{1}{21415755798360000} a^{14} + \frac{317}{214157557983600} a^{12} - \frac{87613}{264653433000} a^{10} - \frac{24763283}{899821672200} a^{8} + \frac{5610428953}{2964118449600} a^{6} + \frac{12915191}{16467324720} a^{4} + \frac{102103997}{242166540} a^{2} + \frac{907675}{4036109}$, $\frac{1}{128494534790160000} a^{15} + \frac{317}{1284945347901600} a^{13} + \frac{1777829}{5398930033200} a^{11} + \frac{141293773}{5398930033200} a^{9} - \frac{125448971}{209231890560} a^{7} - \frac{17220851}{32934649440} a^{5} + \frac{149936669}{726499620} a^{3} + \frac{907675}{24216654} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{156}\times C_{22620}$, which has order $112919040$ (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:  \( 4424425.500726906 \) (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

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{255}) \), \(\Q(\sqrt{102}) \), \(\Q(\sqrt{170}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{10}, \sqrt{102})\), \(\Q(\sqrt{10}, \sqrt{17})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{6}, \sqrt{170})\), \(\Q(\sqrt{15}, \sqrt{17})\), \(\Q(\sqrt{15}, \sqrt{102})\), \(\Q(\sqrt{6}, \sqrt{17})\), 4.0.11319552000.8, 4.0.11319552000.7, 4.0.1257728000.2, 4.0.1257728000.5, 8.8.277102632960000.12, 8.0.128132257480704000000.4, 8.0.1581879721984000000.7, 8.0.512529029922816000000.8, 8.0.512529029922816000000.7, 8.0.512529029922816000000.10, 8.0.512529029922816000000.9

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.24.13$x^{8} + 28 x^{4} + 36$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.13$x^{8} + 28 x^{4} + 36$$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}$
$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}$
$17$17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$