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Properties

Label 16.0.345874011963975936.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 3^{14}\cdot 7^{10}$
Root discriminant $12.48$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $C_4^2:C_2^2.C_2$ (as 16T406)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -155, 446, -837, 1235, -1543, 1642, -1500, 1210, -879, 574, -332, 167, -72, 26, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 26*x^14 - 72*x^13 + 167*x^12 - 332*x^11 + 574*x^10 - 879*x^9 + 1210*x^8 - 1500*x^7 + 1642*x^6 - 1543*x^5 + 1235*x^4 - 837*x^3 + 446*x^2 - 155*x + 25)
 
gp: K = bnfinit(x^16 - 7*x^15 + 26*x^14 - 72*x^13 + 167*x^12 - 332*x^11 + 574*x^10 - 879*x^9 + 1210*x^8 - 1500*x^7 + 1642*x^6 - 1543*x^5 + 1235*x^4 - 837*x^3 + 446*x^2 - 155*x + 25, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 26 x^{14} - 72 x^{13} + 167 x^{12} - 332 x^{11} + 574 x^{10} - 879 x^{9} + 1210 x^{8} - 1500 x^{7} + 1642 x^{6} - 1543 x^{5} + 1235 x^{4} - 837 x^{3} + 446 x^{2} - 155 x + 25 \)

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:  \(345874011963975936=2^{8}\cdot 3^{14}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.48$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{295232710} a^{15} - \frac{8154731}{147616355} a^{14} - \frac{17629927}{147616355} a^{13} - \frac{16027001}{147616355} a^{12} + \frac{64786466}{147616355} a^{11} - \frac{122417147}{295232710} a^{10} + \frac{67698382}{147616355} a^{9} + \frac{47637748}{147616355} a^{8} - \frac{6007831}{59046542} a^{7} + \frac{19499157}{59046542} a^{6} - \frac{113142523}{295232710} a^{5} - \frac{59929789}{147616355} a^{4} + \frac{12278030}{29523271} a^{3} + \frac{143335733}{295232710} a^{2} + \frac{23624813}{147616355} a + \frac{3254624}{29523271}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{50300689}{147616355} a^{15} - \frac{316221303}{147616355} a^{14} + \frac{1088682804}{147616355} a^{13} - \frac{2881577818}{147616355} a^{12} + \frac{6459716358}{147616355} a^{11} - \frac{12378214363}{147616355} a^{10} + \frac{20655621801}{147616355} a^{9} - \frac{30588887611}{147616355} a^{8} + \frac{8156484454}{29523271} a^{7} - \frac{9756984501}{29523271} a^{6} + \frac{50841011843}{147616355} a^{5} - \frac{44729097212}{147616355} a^{4} + \frac{6679999943}{29523271} a^{3} - \frac{20667879188}{147616355} a^{2} + \frac{9185461059}{147616355} a - \frac{383610084}{29523271} \) (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:  \( 311.42757219 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2:C_2^2.C_2$ (as 16T406):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_4^2:C_2^2.C_2$
Character table for $C_4^2:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-7}) \), 4.0.189.1 x2, 4.2.1323.1 x2, \(\Q(\sqrt{-3}, \sqrt{-7})\), 8.0.1750329.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
3Data not computed
$7$7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$