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Properties

Label 16.0.11047269200...5169.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 17^{14}$
Root discriminant $20.66$
Ramified primes $3, 17$
Class number $5$
Class group $[5]$
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 26, -20, 125, -14, 212, -59, 206, -36, 111, -22, 40, -5, 8, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 8*x^14 - 5*x^13 + 40*x^12 - 22*x^11 + 111*x^10 - 36*x^9 + 206*x^8 - 59*x^7 + 212*x^6 - 14*x^5 + 125*x^4 - 20*x^3 + 26*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^16 - x^15 + 8*x^14 - 5*x^13 + 40*x^12 - 22*x^11 + 111*x^10 - 36*x^9 + 206*x^8 - 59*x^7 + 212*x^6 - 14*x^5 + 125*x^4 - 20*x^3 + 26*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 8 x^{14} - 5 x^{13} + 40 x^{12} - 22 x^{11} + 111 x^{10} - 36 x^{9} + 206 x^{8} - 59 x^{7} + 212 x^{6} - 14 x^{5} + 125 x^{4} - 20 x^{3} + 26 x^{2} + 4 x + 1 \)

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:  \(1104726920056229495169=3^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.66$
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:  $3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(51=3\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{51}(1,·)$, $\chi_{51}(2,·)$, $\chi_{51}(4,·)$, $\chi_{51}(8,·)$, $\chi_{51}(13,·)$, $\chi_{51}(16,·)$, $\chi_{51}(19,·)$, $\chi_{51}(25,·)$, $\chi_{51}(26,·)$, $\chi_{51}(32,·)$, $\chi_{51}(35,·)$, $\chi_{51}(38,·)$, $\chi_{51}(43,·)$, $\chi_{51}(47,·)$, $\chi_{51}(49,·)$, $\chi_{51}(50,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{175848618214} a^{15} - \frac{14062920235}{87924309107} a^{14} + \frac{26932135941}{175848618214} a^{13} - \frac{30304037477}{175848618214} a^{12} + \frac{3755248}{1312303121} a^{11} - \frac{10113708245}{175848618214} a^{10} - \frac{45695586599}{175848618214} a^{9} - \frac{38378407638}{87924309107} a^{8} - \frac{25838862707}{175848618214} a^{7} + \frac{54676905473}{175848618214} a^{6} + \frac{212605009}{853634069} a^{5} + \frac{55432699327}{175848618214} a^{4} - \frac{26642787345}{175848618214} a^{3} - \frac{5139250882}{87924309107} a^{2} + \frac{24003914841}{175848618214} a + \frac{35857162949}{87924309107}$

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

Class group and class number

$C_{5}$, which has order $5$

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:  \( -\frac{34167011117}{175848618214} a^{15} + \frac{19411801175}{87924309107} a^{14} - \frac{274127455459}{175848618214} a^{13} + \frac{102130431142}{87924309107} a^{12} - \frac{10155045990}{1312303121} a^{11} + \frac{921272081641}{175848618214} a^{10} - \frac{1876615237234}{87924309107} a^{9} + \frac{837457938480}{87924309107} a^{8} - \frac{6837951005309}{175848618214} a^{7} + \frac{1447475152558}{87924309107} a^{6} - \frac{33373112776}{853634069} a^{5} + \frac{1336584065627}{175848618214} a^{4} - \frac{1885269351946}{87924309107} a^{3} + \frac{719148246400}{87924309107} a^{2} - \frac{703590766513}{175848618214} a + \frac{71091820205}{175848618214} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3640.01221338 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

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 $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-51}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{17})\), 4.4.4913.1, 4.0.44217.1, 8.0.1955143089.1, \(\Q(\zeta_{17})^+\), 8.0.33237432513.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed
17Data not computed