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Properties

Label 16.0.93219557956...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 17^{12}$
Root discriminant $55.99$
Ramified primes $2, 5, 17$
Class number $32$ (GRH)
Class group $[4, 8]$ (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![21381376, 0, -5345344, 0, 1021904, 0, -176868, 0, 29189, 0, -2601, 0, 221, 0, -17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 17*x^14 + 221*x^12 - 2601*x^10 + 29189*x^8 - 176868*x^6 + 1021904*x^4 - 5345344*x^2 + 21381376)
 
gp: K = bnfinit(x^16 - 17*x^14 + 221*x^12 - 2601*x^10 + 29189*x^8 - 176868*x^6 + 1021904*x^4 - 5345344*x^2 + 21381376, 1)
 

Normalized defining polynomial

\( x^{16} - 17 x^{14} + 221 x^{12} - 2601 x^{10} + 29189 x^{8} - 176868 x^{6} + 1021904 x^{4} - 5345344 x^{2} + 21381376 \)

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:  \(9321955795676176000000000000=2^{16}\cdot 5^{12}\cdot 17^{12}\)
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$$\chi_{340}(1,·)$, $\chi_{340}(259,·)$, $\chi_{340}(69,·)$, $\chi_{340}(327,·)$, $\chi_{340}(137,·)$, $\chi_{340}(273,·)$, $\chi_{340}(33,·)$, $\chi_{340}(123,·)$, $\chi_{340}(101,·)$, $\chi_{340}(169,·)$, $\chi_{340}(237,·)$, $\chi_{340}(47,·)$, $\chi_{340}(319,·)$, $\chi_{340}(183,·)$, $\chi_{340}(251,·)$, $\chi_{340}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{17} a^{4}$, $\frac{1}{17} a^{5}$, $\frac{1}{17} a^{6}$, $\frac{1}{17} a^{7}$, $\frac{1}{289} a^{8}$, $\frac{1}{578} a^{9} - \frac{1}{34} a^{7} - \frac{1}{34} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{116756} a^{10} - \frac{1}{1156} a^{8} + \frac{1}{68} a^{6} - \frac{1}{68} a^{4} + \frac{1}{4} a^{2} + \frac{49}{101}$, $\frac{1}{233512} a^{11} - \frac{1}{2312} a^{9} - \frac{3}{136} a^{7} - \frac{1}{136} a^{5} - \frac{3}{8} a^{3} - \frac{26}{101} a$, $\frac{1}{7939408} a^{12} - \frac{1}{467024} a^{10} - \frac{7}{4624} a^{8} - \frac{5}{272} a^{6} + \frac{1}{272} a^{4} - \frac{9}{404} a^{2} + \frac{13}{101}$, $\frac{1}{15878816} a^{13} - \frac{1}{934048} a^{11} - \frac{7}{9248} a^{9} - \frac{5}{544} a^{7} - \frac{15}{544} a^{5} + \frac{395}{808} a^{3} - \frac{44}{101} a$, $\frac{1}{31757632} a^{14} - \frac{1}{31757632} a^{12} - \frac{3}{1868096} a^{10} - \frac{21}{18496} a^{8} - \frac{15}{1088} a^{6} + \frac{251}{27472} a^{4} - \frac{23}{404} a^{2} + \frac{35}{101}$, $\frac{1}{63515264} a^{15} - \frac{1}{63515264} a^{13} - \frac{3}{3736192} a^{11} - \frac{21}{36992} a^{9} + \frac{49}{2176} a^{7} + \frac{251}{54944} a^{5} - \frac{23}{808} a^{3} + \frac{35}{202} a$

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

Class group and class number

$C_{4}\times C_{8}$, which has order $32$ (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:  \( \frac{1}{1984852} a^{12} + \frac{65}{404} a^{2} \) (order $10$)
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:  \( 2001852.022248575 \) (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, 4.0.78608.1, 4.0.1965200.1, \(\Q(\zeta_{5})\), 4.0.36125.1, 8.8.96550276000000.1, 8.0.3862011040000.4, 8.0.1305015625.1

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
5Data not computed
17Data not computed