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Properties

Label 16.16.9321955795...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{16}\cdot 5^{12}\cdot 17^{12}$
Root discriminant $55.99$
Ramified primes $2, 5, 17$
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![956, 6564, -3558, -27422, 8077, 41790, -11572, -29394, 8440, 10314, -3022, -1790, 512, 142, -38, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 38*x^14 + 142*x^13 + 512*x^12 - 1790*x^11 - 3022*x^10 + 10314*x^9 + 8440*x^8 - 29394*x^7 - 11572*x^6 + 41790*x^5 + 8077*x^4 - 27422*x^3 - 3558*x^2 + 6564*x + 956)
 
gp: K = bnfinit(x^16 - 4*x^15 - 38*x^14 + 142*x^13 + 512*x^12 - 1790*x^11 - 3022*x^10 + 10314*x^9 + 8440*x^8 - 29394*x^7 - 11572*x^6 + 41790*x^5 + 8077*x^4 - 27422*x^3 - 3558*x^2 + 6564*x + 956, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 38 x^{14} + 142 x^{13} + 512 x^{12} - 1790 x^{11} - 3022 x^{10} + 10314 x^{9} + 8440 x^{8} - 29394 x^{7} - 11572 x^{6} + 41790 x^{5} + 8077 x^{4} - 27422 x^{3} - 3558 x^{2} + 6564 x + 956 \)

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:  \(9321955795676176000000000000=2^{16}\cdot 5^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.99$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(340=2^{2}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{340}(1,·)$, $\chi_{340}(67,·)$, $\chi_{340}(69,·)$, $\chi_{340}(327,·)$, $\chi_{340}(203,·)$, $\chi_{340}(81,·)$, $\chi_{340}(149,·)$, $\chi_{340}(89,·)$, $\chi_{340}(101,·)$, $\chi_{340}(103,·)$, $\chi_{340}(169,·)$, $\chi_{340}(47,·)$, $\chi_{340}(307,·)$, $\chi_{340}(183,·)$, $\chi_{340}(123,·)$, $\chi_{340}(21,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{28} a^{12} + \frac{1}{7} a^{11} - \frac{1}{4} a^{10} - \frac{1}{14} a^{9} - \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{13}{28} a^{6} - \frac{3}{14} a^{5} + \frac{3}{28} a^{4} - \frac{3}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{28} a^{13} + \frac{5}{28} a^{11} - \frac{1}{14} a^{10} + \frac{1}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{4} a^{7} - \frac{5}{14} a^{6} - \frac{1}{28} a^{5} + \frac{1}{7} a^{4} + \frac{2}{7} a^{3} - \frac{1}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{13245036} a^{14} + \frac{50524}{3311259} a^{13} + \frac{41725}{13245036} a^{12} + \frac{172235}{946074} a^{11} - \frac{360019}{3311259} a^{10} + \frac{67403}{315358} a^{9} + \frac{1251841}{13245036} a^{8} - \frac{2519837}{6622518} a^{7} - \frac{1495631}{4415012} a^{6} - \frac{1445119}{3311259} a^{5} - \frac{205977}{1103753} a^{4} - \frac{2620085}{6622518} a^{3} + \frac{1761841}{6622518} a^{2} - \frac{218246}{473037} a + \frac{123976}{3311259}$, $\frac{1}{14058267965364} a^{15} - \frac{87407}{3514566991341} a^{14} + \frac{118203330935}{7029133982682} a^{13} - \frac{35659240361}{14058267965364} a^{12} - \frac{275233087939}{2008323995052} a^{11} + \frac{728971365545}{4686089321788} a^{10} - \frac{388534866815}{2008323995052} a^{9} - \frac{1460624785271}{7029133982682} a^{8} - \frac{169887417481}{2343044660894} a^{7} + \frac{3361981231205}{14058267965364} a^{6} - \frac{274864313611}{4686089321788} a^{5} + \frac{6156651803699}{14058267965364} a^{4} + \frac{1470101967571}{7029133982682} a^{3} - \frac{1500197024036}{3514566991341} a^{2} + \frac{1565512205458}{3514566991341} a + \frac{1271908286}{4901767073}$

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:  \( 1218421193.9371946 \) (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

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}) \), 4.4.9826000.2, \(\Q(\sqrt{5}, \sqrt{17})\), 4.4.9826000.1, 4.4.4913.1, 4.4.122825.1, \(\Q(\zeta_{20})^+\), 4.4.578000.1, 8.8.96550276000000.1, 8.8.15085980625.1, 8.8.334084000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
5Data not computed
17Data not computed