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Properties

Label 16.16.1024959879...4576.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{44}\cdot 17^{12}$
Root discriminant $56.32$
Ramified primes $2, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![191, 1124, -1282, -10640, 4801, 21352, -7538, -17924, 5686, 7268, -2170, -1432, 405, 128, -34, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 34*x^14 + 128*x^13 + 405*x^12 - 1432*x^11 - 2170*x^10 + 7268*x^9 + 5686*x^8 - 17924*x^7 - 7538*x^6 + 21352*x^5 + 4801*x^4 - 10640*x^3 - 1282*x^2 + 1124*x + 191)
 
gp: K = bnfinit(x^16 - 4*x^15 - 34*x^14 + 128*x^13 + 405*x^12 - 1432*x^11 - 2170*x^10 + 7268*x^9 + 5686*x^8 - 17924*x^7 - 7538*x^6 + 21352*x^5 + 4801*x^4 - 10640*x^3 - 1282*x^2 + 1124*x + 191, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 34 x^{14} + 128 x^{13} + 405 x^{12} - 1432 x^{11} - 2170 x^{10} + 7268 x^{9} + 5686 x^{8} - 17924 x^{7} - 7538 x^{6} + 21352 x^{5} + 4801 x^{4} - 10640 x^{3} - 1282 x^{2} + 1124 x + 191 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10249598790959829536343064576=2^{44}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.32$
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:  $2, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(272=2^{4}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{272}(1,·)$, $\chi_{272}(69,·)$, $\chi_{272}(225,·)$, $\chi_{272}(137,·)$, $\chi_{272}(13,·)$, $\chi_{272}(205,·)$, $\chi_{272}(81,·)$, $\chi_{272}(149,·)$, $\chi_{272}(217,·)$, $\chi_{272}(89,·)$, $\chi_{272}(157,·)$, $\chi_{272}(33,·)$, $\chi_{272}(101,·)$, $\chi_{272}(169,·)$, $\chi_{272}(237,·)$, $\chi_{272}(21,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{24} a^{12} - \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{5}{12} a^{8} - \frac{1}{3} a^{7} - \frac{5}{12} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{24}$, $\frac{1}{24} a^{13} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{5}{12} a^{9} - \frac{1}{3} a^{8} - \frac{5}{12} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{24} a$, $\frac{1}{1829424} a^{14} - \frac{5445}{304904} a^{13} + \frac{1951}{1829424} a^{12} + \frac{69409}{457356} a^{11} + \frac{68863}{304904} a^{10} - \frac{27399}{76226} a^{9} + \frac{4526}{38113} a^{8} - \frac{137207}{457356} a^{7} + \frac{129815}{304904} a^{6} + \frac{27803}{457356} a^{5} - \frac{69823}{457356} a^{4} + \frac{65713}{152452} a^{3} - \frac{301089}{609808} a^{2} - \frac{115799}{914712} a + \frac{105203}{1829424}$, $\frac{1}{74000433136848} a^{15} + \frac{1452493}{37000216568424} a^{14} + \frac{610197250399}{74000433136848} a^{13} - \frac{16907980742}{4625027071053} a^{12} - \frac{8103666770339}{37000216568424} a^{11} + \frac{1588558622719}{9250054142106} a^{10} - \frac{172827195215}{4625027071053} a^{9} - \frac{1730969769799}{6166702761404} a^{8} + \frac{1804446165063}{12333405522808} a^{7} - \frac{3225925917623}{18500108284212} a^{6} + \frac{2615922984041}{18500108284212} a^{5} - \frac{240767435803}{6166702761404} a^{4} - \frac{25437502590323}{74000433136848} a^{3} + \frac{2926304364887}{12333405522808} a^{2} + \frac{32014803845491}{74000433136848} a - \frac{3520212346849}{18500108284212}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 1179476773.631501 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{17}) \), \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{2}, \sqrt{17})\), 4.4.591872.2, 4.4.10061824.1, 4.4.10061824.2, 4.4.4913.1, 4.4.314432.1, 8.8.350312464384.1, 8.8.101240302206976.1, 8.8.98867482624.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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}$
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$