↑

Properties

Label 16.0.10826502450...8273.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 7^{8}\cdot 17^{15}$
Root discriminant $65.26$
Ramified primes $3, 7, 17$
Class number $7492$ (GRH)
Class group $[2, 3746]$ (GRH)
Galois group $C_{16}$ (as 16T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37883311, -31242686, 31242686, -15305186, 15305186, -4148936, 4148936, -642686, 642686, -58311, 58311, -3061, 3061, -86, 86, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 86*x^14 - 86*x^13 + 3061*x^12 - 3061*x^11 + 58311*x^10 - 58311*x^9 + 642686*x^8 - 642686*x^7 + 4148936*x^6 - 4148936*x^5 + 15305186*x^4 - 15305186*x^3 + 31242686*x^2 - 31242686*x + 37883311)
 
gp: K = bnfinit(x^16 - x^15 + 86*x^14 - 86*x^13 + 3061*x^12 - 3061*x^11 + 58311*x^10 - 58311*x^9 + 642686*x^8 - 642686*x^7 + 4148936*x^6 - 4148936*x^5 + 15305186*x^4 - 15305186*x^3 + 31242686*x^2 - 31242686*x + 37883311, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 86 x^{14} - 86 x^{13} + 3061 x^{12} - 3061 x^{11} + 58311 x^{10} - 58311 x^{9} + 642686 x^{8} - 642686 x^{7} + 4148936 x^{6} - 4148936 x^{5} + 15305186 x^{4} - 15305186 x^{3} + 31242686 x^{2} - 31242686 x + 37883311 \)

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:  \(108265024508940221449655688273=3^{8}\cdot 7^{8}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.26$
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:  $3, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(357=3\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{357}(64,·)$, $\chi_{357}(1,·)$, $\chi_{357}(335,·)$, $\chi_{357}(209,·)$, $\chi_{357}(274,·)$, $\chi_{357}(20,·)$, $\chi_{357}(167,·)$, $\chi_{357}(41,·)$, $\chi_{357}(106,·)$, $\chi_{357}(43,·)$, $\chi_{357}(146,·)$, $\chi_{357}(125,·)$, $\chi_{357}(169,·)$, $\chi_{357}(253,·)$, $\chi_{357}(62,·)$, $\chi_{357}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{8275601} a^{9} + \frac{1451671}{8275601} a^{8} + \frac{45}{8275601} a^{7} + \frac{137633}{8275601} a^{6} + \frac{675}{8275601} a^{5} - \frac{2417388}{8275601} a^{4} + \frac{3750}{8275601} a^{3} - \frac{1393951}{8275601} a^{2} + \frac{5625}{8275601} a + \frac{2232131}{8275601}$, $\frac{1}{8275601} a^{10} + \frac{50}{8275601} a^{8} + \frac{1017246}{8275601} a^{7} + \frac{875}{8275601} a^{6} + \frac{2501206}{8275601} a^{5} + \frac{6250}{8275601} a^{4} + \frac{185257}{8275601} a^{3} + \frac{15625}{8275601} a^{2} - \frac{3674658}{8275601} a + \frac{6250}{8275601}$, $\frac{1}{8275601} a^{11} + \frac{2914105}{8275601} a^{8} - \frac{1375}{8275601} a^{7} + \frac{3895157}{8275601} a^{6} - \frac{27500}{8275601} a^{5} - \frac{3079358}{8275601} a^{4} - \frac{171875}{8275601} a^{3} - \frac{181916}{8275601} a^{2} - \frac{275000}{8275601} a - \frac{4023737}{8275601}$, $\frac{1}{8275601} a^{12} - \frac{1650}{8275601} a^{8} - \frac{3105553}{8275601} a^{7} - \frac{38500}{8275601} a^{6} - \frac{507195}{8275601} a^{5} - \frac{309375}{8275601} a^{4} + \frac{3993255}{8275601} a^{3} - \frac{825000}{8275601} a^{2} - \frac{1898781}{8275601} a - \frac{343750}{8275601}$, $\frac{1}{8275601} a^{13} + \frac{502908}{8275601} a^{8} + \frac{35750}{8275601} a^{7} + \frac{3146028}{8275601} a^{6} + \frac{804375}{8275601} a^{5} - \frac{4132864}{8275601} a^{4} - \frac{2913101}{8275601} a^{3} - \frac{1300853}{8275601} a^{2} + \frac{661899}{8275601} a + \frac{373705}{8275601}$, $\frac{1}{8275601} a^{14} + \frac{45500}{8275601} a^{8} - \frac{2933630}{8275601} a^{7} + \frac{1194375}{8275601} a^{6} + \frac{3979478}{8275601} a^{5} + \frac{1961899}{8275601} a^{4} - \frac{368825}{8275601} a^{3} + \frac{3610697}{8275601} a^{2} + \frac{1771747}{8275601} a + \frac{3911899}{8275601}$, $\frac{1}{8275601} a^{15} + \frac{1883052}{8275601} a^{8} - \frac{853125}{8275601} a^{7} - \frac{1967666}{8275601} a^{6} - \frac{3923798}{8275601} a^{5} - \frac{227716}{8275601} a^{4} - \frac{1502283}{8275601} a^{3} + \frac{2336183}{8275601} a^{2} - \frac{3757571}{8275601} a - \frac{3785028}{8275601}$

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

Class group and class number

$C_{2}\times C_{3746}$, which has order $7492$ (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:  \( 3640.012213375973 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{16}$ (as 16T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 16
The 16 conjugacy class representatives for $C_{16}$
Character table for $C_{16}$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R $16$ R $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
3Data not computed
7Data not computed
17Data not computed