↑

Properties

Label 16.0.56056588304...6656.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 13^{8}$
Root discriminant $17.15$
Ramified primes $2, 13$
Class number $3$
Class group $[3]$
Galois group $D_4\times C_2$ (as 16T9)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 50, -144, 226, -136, -174, 340, -21, -380, 318, -28, -56, 0, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 56*x^12 - 28*x^11 + 318*x^10 - 380*x^9 - 21*x^8 + 340*x^7 - 174*x^6 - 136*x^5 + 226*x^4 - 144*x^3 + 50*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 20*x^14 - 56*x^12 - 28*x^11 + 318*x^10 - 380*x^9 - 21*x^8 + 340*x^7 - 174*x^6 - 136*x^5 + 226*x^4 - 144*x^3 + 50*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 20 x^{14} - 56 x^{12} - 28 x^{11} + 318 x^{10} - 380 x^{9} - 21 x^{8} + 340 x^{7} - 174 x^{6} - 136 x^{5} + 226 x^{4} - 144 x^{3} + 50 x^{2} - 8 x + 1 \)

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:  \(56056588304600006656=2^{36}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.15$
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:  $2, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{81} a^{12} - \frac{2}{27} a^{11} - \frac{5}{81} a^{10} - \frac{1}{81} a^{9} - \frac{10}{81} a^{8} + \frac{7}{81} a^{7} + \frac{40}{81} a^{6} - \frac{26}{81} a^{5} - \frac{11}{27} a^{4} + \frac{19}{81} a^{3} - \frac{2}{27} a^{2} - \frac{34}{81} a + \frac{26}{81}$, $\frac{1}{405} a^{13} + \frac{1}{405} a^{12} + \frac{34}{405} a^{11} - \frac{7}{45} a^{10} - \frac{17}{405} a^{9} + \frac{2}{45} a^{8} + \frac{62}{405} a^{7} + \frac{146}{405} a^{6} + \frac{1}{405} a^{5} + \frac{166}{405} a^{4} + \frac{181}{405} a^{3} - \frac{76}{405} a^{2} - \frac{77}{405} a + \frac{74}{405}$, $\frac{1}{405} a^{14} - \frac{2}{405} a^{12} - \frac{22}{405} a^{11} - \frac{49}{405} a^{10} - \frac{13}{81} a^{9} - \frac{11}{405} a^{8} - \frac{26}{405} a^{7} - \frac{13}{27} a^{6} - \frac{1}{81} a^{5} + \frac{2}{9} a^{4} + \frac{158}{405} a^{3} - \frac{196}{405} a^{2} - \frac{1}{45} a - \frac{13}{135}$, $\frac{1}{6075} a^{15} + \frac{1}{1215} a^{13} + \frac{2}{1215} a^{12} + \frac{283}{2025} a^{11} - \frac{901}{6075} a^{10} - \frac{58}{1215} a^{9} - \frac{1}{405} a^{8} - \frac{74}{675} a^{7} - \frac{2708}{6075} a^{6} + \frac{1607}{6075} a^{5} + \frac{32}{135} a^{4} + \frac{466}{6075} a^{3} - \frac{691}{6075} a^{2} + \frac{964}{2025} a - \frac{452}{6075}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( \frac{1046}{6075} a^{15} - \frac{523}{405} a^{14} + \frac{3491}{1215} a^{13} + \frac{221}{243} a^{12} - \frac{16627}{2025} a^{11} - \frac{48191}{6075} a^{10} + \frac{57757}{1215} a^{9} - \frac{6356}{135} a^{8} - \frac{5749}{675} a^{7} + \frac{271682}{6075} a^{6} - \frac{118928}{6075} a^{5} - \frac{8149}{405} a^{4} + \frac{182426}{6075} a^{3} - \frac{105491}{6075} a^{2} + \frac{1531}{225} a - \frac{8962}{6075} \) (order $8$)
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:  \( 4427.02461923 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4$ (as 16T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-13}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{2}, \sqrt{-13})\), \(\Q(i, \sqrt{13})\), \(\Q(\sqrt{-2}, \sqrt{-13})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{2}, \sqrt{13})\), \(\Q(i, \sqrt{26})\), \(\Q(\sqrt{-2}, \sqrt{13})\), 4.0.832.1 x2, 4.0.3328.1 x2, 4.2.2704.1 x2, 4.2.10816.1 x2, 8.0.1871773696.1, 8.0.116985856.1, 8.0.1871773696.3, 8.0.44302336.1 x2, 8.4.1871773696.1 x2, 8.0.7487094784.1 x2, 8.0.1871773696.4 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$